![2 1 Preliminaries
Definition 1.1 (Euler [1]) The gamma function due to Euler is defined as
Γ (z) =
∞
0
e−t
tz−1
dt, (1.1)
where Re (z) 0 and z ∈ C.
The formula was discovered by Euler in 1729 (see [1], p.1), and the notation
Γ (z) was introduced by Legendre in 1814 (see [2], p.476).
Theorem 1.1 (Weierstrassian Product [3]) If z ∈ CZ−
0 with Z−
0 =:
{0, −1, −2, · · ·} and γ := lim
n→∞
n
k=1
1
k − log n
is the Euler constant, the Gamma
function was given as [3]
Γ (z) =
e−γ z
z
∞
k=1
1 +
z
k
−1
e
z
k
. (1.2)
Moreover, Γ (z) is analytic except at the points z ∈ Z−
0 , where it has simple
poles [4].
The formula for the Weierstrassian product was discovered by Weierstrass in 1856
[3] and by Newman in 1848 [5], respectively, and the proofs were published by
Hölder [6], Moore [7], and Baines [8].
Definition 1.2 (Euler [1]) Let Re (z) 0 and z ∈ C. Then the Euler’s functional
equation states
Γ (z + 1) = zΓ (z) . (1.3)
The result is the Euler’s functional equation discovered by Euler in 1729 [9] and
reported by Weierstrass [3], Brunel [10], Gronwall [11], and Olver [12].
Theorem 1.2 (Euler [1]) If z ∈ N0, then we have
Γ (z + 1) = z!. (1.4)
The result is the Euler’s functional equation discovered by Euler in 1729 [1, 2] and
discussed by Weierstrass [4], Brunel [10], and Gronwall [11].
Theorem 1.3 (Euler [1])
Γ
1
2
=
√
π. (1.5)
This work was discovered by Euler in 1729 [11] and discussed in Bell [13], Luke
[14], and Bendersky [15].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-24-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 3
Theorem 1.4 (Euler) If n, j ∈ N, then we have
n
j=1
Γ
1 − j
n
= (2π)
n−1
2 n
1
2 . (1.6)
The result was reviewed by Gronwall in 1916 [11].
Theorem 1.5 (Winckler [16]) If z ∈ C and g, k, j, k, l, m, n ∈ N, then we have
n−1
j=0
Γ
hz + hj
n
m−1
l=0
Γ
gz + gl
m
=
h
g
hgz+ hg−h−g
2
(2π)
h−g
2 . (1.7)
The result was discovered by Winckler in 1856 [16] and reviewed by Gronwall in
1916 [11].
Theorem 1.6 (Schlömilch [17] and Newman [5]) If z ∈ C and k ∈ N, then we
have
1
Γ (z)
= eγ z
z
∞
k=1
1 +
z
k
e− z
k . (1.8)
The result was discovered by Schlömilch in 1844 [17] and by Newman [18].
Theorem 1.7 (Whittaker [19]) If Re (z) 0, z ∈ C, and k ∈ N, then we have
∞
0
e−kt
tz−1
dt =
Γ (z)
kz
. (1.9)
The result was first reported by Whittaker in 1902 (see [19], p.184) and further
reported by Whittaker and Watson in 1920 [20].
Theorem 1.8 (Whittaker [19]) If α, β ∈ C, Re (α) 0, and Re (β) 0, then we
have
π
2
0
cosα−1
t sinβ−1
tdt =
1
2
Γ α
2 Γ α
2
Γ
α+β
2
. (1.10)
The result was first defined by Whittaker in 1902 (see [19], p.191) and further
reported by Whittaker and Watson in 1920 [20].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-25-320.jpg)
![4 1 Preliminaries
Theorem 1.9 (Titchmarsh [21]) If α, β ∈ C, Re (α) 0, Re (β) 0, and
Re (α + β) 1, then we have
∞
−∞
1
Γ (α + t) Γ (β − t)
dt =
2α+β−1
Γ (α + β − 1)
. (1.11)
The result was first reported in the Titchmarsh’s monograph [21].
Theorem 1.10 (Titchmarsh [21]) If Re (α) −1, Re (β) −1, and
Re (α + β) −1, then we have
f
∞
k=1
k (α + β + k)
(α + k) (β + k)
=
Γ (α + 1) Γ (β + 1)
Γ (α + β + 1)
. (1.12)
The result was first presented in the Titchmarsh’s monograph [21].
Theorem 1.11 (Titchmarsh [21]) If z ∈ C and k, n ∈ N, then we have
n
k=1
1 −
z
kn
= −
n
k=1
Γ
−e2πi k−1
n z
1
n
−1
. (1.13)
The result was first reported in the Titchmarsh’s monograph [21].
Theorem 1.12 (Euler [22]) Let z ∈ C and Re (z) 0. Then we have the Euler’s
completion formula as follows:
Γ (z) Γ (1 − z) =
π
sin (πz)
(1.14)
and
sin (πz) = πz
n
k=1
1 −
z2
k2
. (1.15)
The result is the Euler’s completion formula due to Euler [22].
For more details of the results, readers refer to Weierstrass [4], Manocha and
Srivastava [23], Luke [14], Bell [13], Godefroy [24] and Tannery [25].
Theorem 1.13 (Legendre [2], p.485)
The Legendre duplication formula states
Γ (2z) Γ
1
2
= 22z−1
Γ (z) Γ
z +
1
2
, (1.16)
where z ∈ CZ−
0 .](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-26-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 5
The Legendre’s duplication formula was first discovered by Legendre in 1809 (see
[2], p.477). For more details of the Legendre duplication formula, readers refer to
Gronwall [11], Andrews et al. [26], and Manocha and Srivastava [23].
Theorem 1.14 (Gauss [27]) If z ∈ C
0, − j
m
with j m and j, m ∈ N, then
we have
Γ (mz) = (2π)
1−m
2 mmz− 1
2
m
j=1
Γ
z +
j − 1
m
. (1.17)
The result is the Gauss’ multiplication formula due to Gauss [27]. For more details
of the Gauss’ multiplication formula, readers refer to Winckler [28], Gronwall [11],
Manocha and Srivastava [23], and Andrews et al. [26].
Theorem 1.15 (Weierstrass [3]) If z ∈ C/Z−
0 , then we have
Γ
1
2 − z
Γ
1
2 +z
= πsec (πz)
= π
cos(πz)
= 2π
eiπz+e−iπz .
(1.18)
The result was discovered by Weierstrass [4] and reported by Bell in 1968 [13] and
by Luke in 1969 [14].
Theorem 1.16 Γ
1
2 + iz
Γ
1
2 − iz
= π
cosh(πz) = 2π
eπz+e−πz , (19)
Γ
1
2
+ iz
2
=
2π
eπz + e−πz
(1.19)
and
Γ (iz) Γ (−iz) =
π
−iz sin (πzi)
=
2π
z eπz − e−πz
, (1.20)
where |z| → ∞.
The results were reported by different researchers, for example, Lerch [29],
Godefroy [30], Stieltjes [31], Bateman [32], and Andrews et al. [26].
Theorem 1.17
n−1
j=1
Γ
j
n
Γ
1 −
j
n
=
(2π)n−1
n
, (1.21)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-27-320.jpg)
![6 1 Preliminaries
Γ
−n +
1
2
= (−1)n 2n√
π
(2n − 1)!
, (1.22)
Γ
n +
1
2
=
(2n − 1)!
√
π
2n
, (1.23)
Γ (n + z) Γ (n − z) =
πz
sin (πz)
((n − 1)!)2
n−1
j=1
Γ
1 −
z2
j
(1.24)
and
Γ
n +
1
2
+ z
Γ
n +
1
2
− z
=
Γ
n + 1
2
2
cos (πz)
n
j=1
Γ
1 −
4z2
(2j − 1)2
,
(1.25)
where n ∈ N and z ∈ CZ−
0 .
The results were reported by Weierstrass in 1856 [4].
Let us introduce the Temme function which is related to the ratio of two gamma
functions [35].
Definition 1.3 The Temme function is defined as
Γ ∗
(z) =
Γ (z)
√
2πzz− 1
2 e−z
, (1.26)
where z ∈ C and Re (z) 0.
The result was defined by in Temme’s book (see [35], p.66).
Theorem 1.18 Let z, a, b ∈ C, Re (z) 0, Re (z + a) 0, and Re (z + b) 0.
Then the ratio of two gamma functions is shown as follows:
Γ (z + a)
Γ (z + b)
= zα−b Γ ∗ (z + a)
Γ ∗ (z + b)
Q (z, a, b) , (1.27)
where
Q (z, a, b) =
1 +
a
z
a− 1
2
1 +
b
z
1
2 −b
e
z
ln
1+ a
z
− a
z −ln
1+ b
z
+ b
z
. (1.28)
The result was discovered by Temme in the book (see [35], p.67).](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-28-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 7
In an alternative manner, we have (see [35], p.67)
Γ (z + a)
Γ (z + b)
=
1
Γ (b − a)
1
0
tz+a−1
(1 − t)b−a−1
dt, (1.29)
where a, b, z ∈ C and Re (b − a) 0.
The result was reported by Temme in the book (see [35], p.67).
Here, we introduce the interested formula reported in the book (see [35], p.72)
as follows:
Theorem 1.19 Let a, b, z ∈ C , Re (a) 0, Re (b) 0, and Re (z) 0.
Then we have
∞
0
tz−1
e−atb
dt =
1
b
Γ
z
b
a− z
b . (1.30)
The result was reported by Temme (see [35], p.72).
There are some special cases of (1.29) as follows:
∞
0
tz−1
e−at
dt = a−z
Γ (z) , (1.31)
∞
0
e−tb
dt = Γ
1
b
+ 1
, (1.32)
∞
0
e−atb
dt =
1
b
Γ
z
b
a− z
b , (1.33)
∞
0
e−at2
dt =
1
2
Γ
z
2
a− z
2 , (1.34)
∞
0
tz−1
e−at2
dt =
1
2
Γ
z
2
a− z
2 , (1.35)
∞
0
tz−1
e−t2
dt =
1
2
Γ
z
2
(1.36)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-29-320.jpg)
![8 1 Preliminaries
and
∞
0
tz−1
e−tb
dt =
1
b
Γ
z
b
, (1.37)
where a, b, z ∈ C , Re (a) 0, Re (b) 0, and Re (z) 0.
There are useful formulas as follows [35]:
Γ
n+
1
2
=
(2n)!
√
π
22nn!
, (1.38)
Γ
−n+
1
2
= (−1)n √
π22n n!
(2n)!
, (1.39)
Γ
1
2
− z
Γ
z+
1
2
=
π
cos (πz)
, z −
1
2
/
∈ Z, (1.40)
Γ
1
2
− xi
Γ
xi+
1
2
=
π
cosh (πx)
, (1.41)
∞
0
tz−1
sin tdt = Γ (z) sin
πz
2
(1.42)
and
∞
0
tz−1
cos tdt = Γ (z) cos
πz
2
, (1.43)
where z ∈ C , Re (z) 0, n ∈ N and x ≥ 0.
Making use of (1.30) and taking a = i =
√
−1, we have
∞
0
tz−1
e−it
dt = i−z
Γ (z) , (1.44)
∞
0
tz−1e−itdt =
∞
0
tz−1 (cos t − i sin t) dt
=
∞
0
tz−1 cos tdt − i
∞
0
tz−1 sin tdt
(1.45)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-30-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 9
and
i−zΓ (z) = e− zπi
2 Γ (z)
= cos zπ
2 − sin zπ
2 Γ (z) ,
(1.46)
where (iθ)z
= |θ| e− zπisgn(z)
2 , z ∈ C and Re (z) 0, such that we obtain (1.42) and
(1.43).
1.1.2 The Pochhammer Symbols and Related Formulas
We now introduce the Pochhammer symbols and related theorems.
Definition 1.4 (Pochhammer [34])
The Pochhammer symbol is defined as [34]
(α)k =
k
n=1
(α + n − 1)
= Γ (α+k)
Γ (α)
=
1 (k = 0)
α (α + 1) · · · (α + k − 1) (k ∈ N0)
(1.47)
and
(α)0 = 1, (1.48)
where α ∈ C and k, n ∈ N.
The Pochhammer symbol was first suggested by Pochhammer in 1870 [34].
The notation was first used by Pochhammer in 1870 [34] and Weierstrass noticed
in 1856 that [4]
Γ (α + k) = α (α + 1) · · · (α + k − 1) Γ (α) (k ∈ N0) . (1.49)
For more information, readers may refer to the monograph [33].
Moreover, there is (see[4, 11, 36])
lim
k→∞
(α)k =
1
Γ (α)
, (1.50)
where α ∈ CC−
0 and k ∈ N.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-31-320.jpg)
![10 1 Preliminaries
Suppose that α = −n and n ∈ N0, then there is (see [36], p.3)
(α)k =
(−n)k , n ≥ k
−, n k.
(1.51)
Theorem 1.20 (Euler [1]) If z ∈ CZ−
0 , then we have
Γ (z) = lim
n→∞
nz
(z)n+1
. (1.52)
The result was discovered by Euler in 1729 [9], reported by Weierstrass in 1856 [3],
and discussed by Gronwall in 1916 [11].
Theorem 1.21 There exist
(α)k (α + k)n = (α)n+k , (1.53)
and
(α + k)m−k =
(α)m
(α)k
, (1.54)
where α ∈ CZ−
0 and n, k ∈ N.
The first formula of the results was reported by Rainville in 1960 (see [37], p.59)
and the second formula was suggested by Slater in 1966 (see [36], p.31).
There are some useful formulas as follows:
(−z)n = (−1)n
(z − n + 1) n, (1.55)
(z)2n = 22n
z
2
n
z
2
+
1
2
n, (1.56)
(z)2n+1 = 22n+1
z
2
n+1
z
2
+
1
2
n, (1.57)
where z ∈ C and n ∈ N.
Theorem 1.22 Let j, k, m, n ∈ N0, k ≤ n and α, β ∈ CZ, then we have
(1)n = n!, (1.58)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-32-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 11
α
n
= α(α−1)···(α−k−1)
n!
= Γ (α+1)
n!Γ (α−n+1)
= 1
n!(α+1)n
= (−1)n(−α)n
n! ,
(1.59)
Γ (α + 1)
Γ (α − n + 1)
= (−1)n
(−α)n , (1.60)
1
(m − n)!
=
(−1)n
(−m)n
m!
, (1.61)
Γ (α − n)
Γ (α)
=
(−1)n
(1 − α)n
, (1.62)
Γ (α − n)
Γ (α)
= (α)−n =
(−1)n
(1 − α)n
, (1.63)
n!
(α)n+1
−
n!
(α + 1)n+1
=
(n + 1)!
(α)n+2
, (1.64)
(α)n
(β)n
−
(α)n+1
(β)n+1
=
(α)n
(β)n+1
(α − β) , (1.65)
(α)n−k =
(−1)k
(1 − α)−n (1 − α − n)k
=
(−1)k
(α)n
(1 − α − n)k
, (1.66)
(1)n−k = (n − k)! =
(−1)k
(1)n
(−n)k
= =
(−1)k
n!
(−n)k
, (1.67)
(α)mn = mmn
m
j=1
α + j − 1
m
n
, (1.68)
and
(−n)k =
(−1)kn!
(n−k)! , (0 ≤ k ≤ n)
0, (k n)
. (1.69)
For more details of the results, readers refer to the works [3, 11, 13, 14, 23, 36].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-33-320.jpg)
![12 1 Preliminaries
Theorem 1.23 (Stirling [38])
1
z−α
=
∞
k=0
(α)k
(z)k+1
=
∞
k=0
Γ (z)
Γ (α)
Γ (α+k)
Γ (z+k+1)
= 1
z + α
z(z+1) + α(α+1)
z(z+1)(z+2) + · · · +,
(1.70)
where Re (α) 0, Re (z) 0, Re (α − z) 0and α, z ∈ CZ−
0 .
The result was discovered by Stirling in 1730 [38] and reviewed by Gronwall in
1916 [11].
Theorem 1.24 Let h(1) (t) 0, h (0) = a, h (1) = b, x ∈ C and Re (x) 0.
Then we have
Γ (x) =
b
a
e−h(t)
(h (t))x−1
h(1) (t) dt . (1.71)
The result was discovered by Yang et al. in 2020 when x ∈ N [39].
1.1.3 The Euler Beta Function
In this section, we investigate the concept and theorems of the Euler beta function.
Definition 1.5 (Euler [22])
The Euler beta function is defined as
B (α, β) =
Γ (α) Γ (β)
Γ (α + β)
=
1
0
tα−1
(1 − t)β−1
dt, (1.72)
where Re (α) 0, Re (β) 0, and α, β ∈ CZ−
0 .
The formula (called the Euler integral of the second kind) was first discovered by
Euler in 1772 [22] and by Legendre in 1811 (see [40], p.211), and the name of
the beta function was introduced for the first time by Binet in 1839 [41]. For more
details, see the monograph [33].
It is clear that
B (α, 1) =
1
α + 1
(1.73)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-34-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 13
and
B (1, β) =
1
1 + β
, (1.74)
where Re (α) 0, Re (β) 0, and α, β ∈ CZ−
0 .
There are some useful formulas as follows:
π
2
0
sin t2α−1
cos2β−1
dt = B (α, β) , (1.75)
b
a
(b − t)α−1
(t − a)β−1
dt = B (α, β) (b − a)α+β−1
, (1.76)
where α, β ∈ CZ−
0 , a, b ∈ R, Re (α) 0, Re (β) 0 and b a.
In the special case of a = 0 and b = x in (1.76), we have
x
0
(x − t)α−1
tβ−1
dt = B (α, β) xα+β−1
, (1.77)
where α, β ∈ CZ−
0 , x 0 and Re (α) 0, Re (β) 0 x 0.
Theorem 1.25 (Whittaker and Watson [20])
Γ (α, n) = lim
n→∞
nα
B (α, n) , (1.78)
where α ∈ CZ−
0 and n ∈ N.
The result was reported by Whittaker and Watson (see [20], p.254).
Theorem 1.26 (Euler [22])
B (α, β) (α + β, c) = B (β, c) (β + c, α) . (1.79)
The result was discovered by Euler in 1772 [22] and further reported by Whittaker
and Watson (see [20], p.261).
Theorem 1.27 (Whittaker and Watson [20])
B (α, β) =
∞
0
tα−1
(1 + t)−(α+β)
dt, (1.80)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-35-320.jpg)
![14 1 Preliminaries
B (α, n + 1) =
n!
(α)n+1
(n ∈ N0) , (1.81)
B (α, β + 1) =
β
α + β
B (α, β) , (1.82)
B (α, β) = B (α + 1, β) + B (α, β + 1) , (1.83)
B (α, β + 1) =
β
α
B (α + 1, β) , (1.84)
and
B (α, β) = B (β, α) . (1.85)
The results were discovered by Whittaker and Watson (see [20], p.254).
Theorem 1.28 (Whittaker and Watson [20])
There is [20]
B (nα, nβ) = n−nβ
n
k=1
B
β + n−1
n , α
n
k=1
B ((k − 1) α, α)
, (1.86)
where Re (α) 0, Re (β) 0, n ∈ N and α, β ∈ CZ−
0 .
Theorem 1.29 (Hankel [42])
1
Γ (z)
=
1
2πi
L
s−z
es
ds, s ∈ C, (1.87)
where L is the Hankel contour.
The result was discovered by Hankel in 1864 [42].
There is an alternative integral representation as follows:
Theorem 1.30 (Hankel [42])
1
Γ (z)
=
i
2π
ℵ
(−s)−z
e−s
ds, s ∈ C, (1.88)
where ℵ is the loop contour, which starts at 0i + ∞ encircles the origin and tends
to 0i − ∞.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-36-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 15
The result was discovered by Hankel in 1864 [42].
Definition 1.6
ψ (z) =
d log Γ (z)
dz
=
Γ (1) (z)
Γ (z)
, (1.89)
where s ∈ C and
log Γ (z) =
z −
1
2
log z − s + log
√
2π + Σ (z) (1.90)
with the infinite series due to Gudermann [43] Σ (z).
The result is the logarithmic derivative of the gamma function [11].
For more information of the Euler gamma function, Pochhammer symbols, and
Euler beta function, reader refers to Gronwall [11], Bell [13], Luke [14], Whittaker
and Watson [20], Manocha and Srivastava [23], Wang et al. [35], Slater [36],
Rainville [37], and Andrews et al. [26].
Theorem 1.31 Let h(1) (t) 0, h (0) = a, h (∞) = b, x, y ∈ C , Re (x) 0 and
Re (y) 0.
Then we have
B (x, y) =
b
a
e−h(t) (h (t))x−1
(1 − h (t))y−1
h(1) (t) dt . (1.91)
The result was discovered by Yang et al. in 2020 when x ∈∈ R+ and y ∈ R+ [39].
1.1.4 The Extended Euler Gamma Functions
In this section, we preset the Euler–Chaudhry–Zubair gamma function, which is the
extension of the gamma function, and other versions of the gamma-type functions.
Definition 1.7 (Chaudhry and Zubair [44])
The Euler–Chaudhry–Zubair gamma function, denoted byI Γp (z), is defined as
I Γp (z) =
∞
0
tz−1
e−t− p
t dt, (1.92)
where Re (z) 0, Re (p) 0, and z ∈ C.
The result was discovered by Chaudhry and Zubair in 1994 [44].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-37-320.jpg)
![18 1 Preliminaries
Definition 1.13 The Euler-type gamma function, denoted by V II Γp,r (z), is defined
as
V II Γp,r (z) =
∞
0
tz−1
e−rtp
dt, (1.109)
where Re (z) 0, Re (p) 0, Re (r) 0, and r, z, p ∈ C.
Theorem 1.34 If Re (z) 0, Re (p) 0, Re (r) 0, and r, z, p ∈ C, then we
have
V II Γp,r (z) =
r
− z
q
p
Γ
z
p
.
Corollary 1.1 If Re (z) 0, Re (r) 0, and z, r ∈ C, then we have
V II Γ1,r (z) = r−z
Γ (z) . (1.110)
Corollary 1.2 If Re (z) 0, Re (p) 0, and z, p ∈ C, then we have
V II Γp,1 (z) =
1
p
Γ
z
p
. (1.111)
The results can be obtained from the analytic number theory in Chap. 2.
For more details, see [9, 45–47].
1.1.5 The Extended Euler Beta Functions
In this section, we preset the Euler–Chaudhry–Qadir–Rafique–Zubair beta function
[48], which is the extension of the beta function, and other related functions.
Definition 1.14 The Euler-type beta function, denoted by I Bp (α, β), is defined as
I Bp (α, β) =
1
0
tα−1
(1 − t)β−1
e− p
t dt, (1.112)
where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 .](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-40-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 19
Definition 1.15 (Chaudhry et al. [48])
The Euler–Chaudhry–Qadir–Rafique–Zubair beta function, denoted by
II Bp (α, β), is defined as [48]
II Bp (α, β) =
1
0
tα−1
(1 − t)β−1
e− p
t(1−t) dt, (1.113)
where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 .
The result was discovered by Chaudhry et al. in 1997 [48].
In this case, we take the form
II Bp (α, β)
=
1
0
tα−1 (1 − t)β−1
e− p
t(1−t) dt
=
1
0
tα−1
(1 − t)β−1
e− p
1−t − p
t dt,
(1.114)
where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 .
Definition 1.16 The Euler-type beta function, denoted by III Bp (α, β), is defined
as
III Bp (α, β) =
1
0
tα−1
(1 − t)β−1
e− p
1−t dt, (1.115)
where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 .
Definition 1.17 The Euler–Choi–Rathie–Parmar beta function, denoted by
IV Bp,q (α, β), is defined as [49]
IV Bp,q (α, β) =
1
0
tα−1
(1 − t)β−1
e− p
t − q
1−t dt, (1.116)
where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, and α, β ∈ CZ−
0 .
The result was discovered by Choi et al. in 2014 [49].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-41-320.jpg)
![20 1 Preliminaries
Definition 1.18 The Euler-type beta function, denoted by V Bp (α, β), is defined as
V Bp (α, β) =
1
0
tα−1
(1 − t)β−1
e−pt
dt, (1.117)
where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 .
Definition 1.19 The Euler-type beta function, denoted by V I Bp,q,r (α, β), is
defined as
V I Bp,q,r (α, β) =
1
0
tα−1
(1 − t)β−1
e−rt− p
t − q
1−t dt, (1.118)
where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, Re (r) 0, and α, β ∈
CZ−
0 .
Theorem 1.35 Let Re (β) Re (α) 0, Re (p) 0, Re (q) 0, Re (r) 0
and α, β ∈ CZ−
0 . Then we have the following:
V I B0,0,p (α, β) = V Bp (α, β) , (1.119)
V I Bp,q,0 (α, β) = IV Bp,q (α, β) , (1.120)
V I B0,q,0 (α, β) = III Bp (α, β) , (1.121)
V I Bp,p,0 (α, β) = II Bp (α, β) , (1.122)
V I Bp,0,0 (α, β) = I Bp (α, β) . (1.123)
Theorem 1.36 (Chaudhry et al. [50])
Suppose that Re (β) Re (α) 0, Re (γ ) 0, |arg (1 − z)| π, Re (p) 0,
and α, β ∈ CZ−
0 , then we have the following integral representation [50]
Fp (γ, α, β; z) =
1
B (α, β − α)
1
0
tα−1
(1 − t)β−α−1
(1 − zt)−γ
e− p
t(1−t) dt.
(1.124)
The result was discovered by Chaudhry et al. in 2004 [50].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-42-320.jpg)
![1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 21
Theorem 1.37 (Chaudhry–Qadir–Srivastava–Paris Theorem [50])
Suppose that Re (β) Re (α) 0, Re (p) 0, Re (z) 0, and α, β ∈ CZ−
0 ,
then we have the following integral representation [50]
Φp (α, β; z) =
1
B (α, β − α)
1
0
tα−1
(1 − t)β−α−1
e
zt− p
t(1−t)
dt. (1.125)
The result was discovered by Chaudhry et al in 2004 [50]
As the direct results, we have the following:
Corollary 1.3 Suppose that Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 ,
then we have the following integral representation
Φ0 (α, β; z) =
1
B (α, β − α)
1
0
tα−1
(1 − t)β−α−1
ezt
dt. (1.126)
Proof For the above formula with p = 0, we get the result.
Corollary 1.4 Suppose that Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ−
0 ,
then we have
Φ0 (α, β; z) =
V Bp (α,)
B (α, β − α)
. (1.127)
Proof With the definition, we directly obtain the result.
Corollary 1.5 Suppose that Re (β) Re (α) 0, Re (p) 0, Re (z) 0, and
α, β ∈ CZ−
0 , then we have the following integral representation
Φp (α, β; 0) =
II Bp (α, β)
B (α, β − α)
. (1.128)
Proof For z = 0, we have the result.
Theorem 1.38 (Choi et al. [49])
Suppose that Re (β) Re (α) 0, p 0, q 0, and α, β ∈ CZ−
0 , then the
integral representation
1
0
1
0
IV Bp,q (α, β) pκ−1
qυ−1
dpdq = Γ (κ) Γ (υ) B (α + κ, β + υ) (1.129)
holds true.
The result was discovered by Choi et al in 2014 [49].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-43-320.jpg)
![22 1 Preliminaries
There are the connections among them as follows:
Theorem 1.39 If Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C, then we have
the following relationships:
V I B0,0,1 (z, 0) = B (z, 0) , (1.130)
V I Bp,q,1 (z, 0) = IV Bp,q (z, 0) , (1.131)
V I B0,0,p (z, 0) = V Bp (z, 0) , (1.132)
V I B0,p,1 (z, 0) = III Bp (z, 0) , (1.133)
V I Bp,p,1 (z, 0) = II Bp (z, 0) , (1.134)
V I Bp,0,1 (z, 0) = I Bp (z, 0) , (1.135)
where Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C.
Definition 1.20 The Euler-type beta function, denoted by V II Bp,r (α, β), is defined
as
V II Bp,r (α, β) =
1
0
tα−1
(1 − t)β−1
e−rtp
dt, (1.136)
where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, and α, β ∈ CZ−
0 .
For the detailed comments of the above formulae, see [45–48].
1.2 Gauss Hypergeometric Series and Supertrigonometric
and Superhyperbolic Functions
In this section, we present the definition, theorems, and properties for the Gauss
hypergeometric series, and the relationships among the hypergeometric super-
trigonometric and superhyperbolic functions via Gauss hypergeometric series.
1.2.1 The Gauss Hypergeometric Series
Let us start with the definition of the Gauss hypergeometric series.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-44-320.jpg)
![1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 23
Definition 1.21 (Gauss [27])
The Gauss hypergeometric series of form is defined as [27]
2F1 (a, b; c; z)
= 1+ab
c z + a(a+1)b(b+1)
c(c+1)
z2
2 + · · ·
=
∞
n=0
(a)n(b)n
(c)n
zn
n! ,
(1.137)
where a, b, c, z ∈ C, n ∈ N0, and |z| 1.
The formula was discovered by Gauss in 1812 [27].
Theorem 1.40 (Euler [51])
If a, b, c, z ∈ C and |z| 1, then there exists
2F1 (a, b; c; z) =
Γ (c)
Γ (b) Γ (c − b)
1
0
tb−1
(1 − t)c−b−1
(1 − zt)−a
dt, (1.138)
where Re (c) Re (b) 0.
The result was discovered by Euler in 1679 [51].
Theorem 1.41 (Gauss [27])
If Re (c − a − b) 0, Re (c) Re (a), and Re (c) Re (b), then there exist
2F1 (a, b; c; 1) =
∞
n=0
(a)n (b)n
(c)n
1
n!
=
Γ (c) Γ (c − b − a)
Γ (c − a) Γ (c − b)
. (1.139)
For |z| 1 and a, b, c, z ∈ C the function 2F1 (a, b; c; z) is analytic except for
simple poles at c = 0 and c ∈ Z−.
The result was discovered by Gauss in 1812 [27].
Theorem 1.42 (Chu-Vandermonde Identity [52, 53])
2F1 (−n, b; c; 1) =
(c − b)n
(c)n
. (1.140)
The result was discovered by Chu in 1303 [52] and by Vandermonde in 1772 [53],
respectively.
Theorem 1.43 (Bateman [54])
If Re (c) Re (μ) 0 and |z| 1, then there is [54]
2F1 (a, b; c + μ; z) =
Γ (c + μ) z1−(c+μ)
Γ (c) Γ (μ)
z
0
tc−1
(z − t)μ−1
2F1 (a, b; c, t) dt.
(1.141)
The result was discovered by Bateman in 1909 [54].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-45-320.jpg)
![24 1 Preliminaries
Theorem 1.44 (Bateman [54])
If Re (c) Re (μ) 0 and |z| 1, then there is [54]
2F1 (a, b; c + μ; z) =
Γ (c + μ)
Γ (c) Γ (μ)
1
0
tc−1
(1 − t)μ−1
2F1 (a, b; c, zt) dt.
(1.142)
This is the special case of Bateman [54].
Theorem 1.45 (Koshliakov [55])
If Re (c) Re (μ) 0 and |z| 1, then there is
1F1 (a, b; c + μ; z) =
Γ (c + μ)
Γ (c) Γ (μ)
1
0
tc−1
(1 − t)μ−1
1F1 (a, b; c, zt) dt.
(1.143)
This result is known as the Koshliakov’s formula [55].
Property 1.1 The hypergeometric series is a solution of the Gauss differential
equation of form
z (1 − z)
d2ϕ
dz2
+ (c − (a + b + 1) z)
dϕ
dz
− abϕ=0, (1.144)
where a, b, z ∈ C , c ∈ CZ−
0 , and |z| 1.
The result was reported by Whittaker and Watson in 1927 [20], and by Bailey in
1935 [56].
Theorem 1.46 (Barnes [57])
Suppose that Re (c) Re (μ) 0, |z| 1, and |arg (−z)| π, then we have
[57]
2F1 (a, b; c; z) =
Γ (c)
Γ (a) Γ (b)
1
2πi
i∞
−i∞
Γ (a + s) Γ (b + s) Γ (−s)
Γ (c + s)
(−z)s
ds.
(1.145)
The result was reported by Barnes in 1908 [57].
Some applications of the ODEs in mechanics of shells were presented by Sun in
[58–61].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-46-320.jpg)
![1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 25
1.2.2 The Hypergeometric Supertrigonometric Functions via
Gauss Superhyperbolic Series
We now consider the hypergeometric supertrigonometric functions via Gauss
superhyperbolic series [39].
Definition 1.22 The hypergeometric supersine via Gauss superhyperbolic series is
defined as [39]
2Supersin1 (a, b; c; z) =
∞
n=0
(a)2n+1 (b)2n+1
(c)2n+1
(−1)n
z2n+1
(2n + 1)!
, (1.146)
where a, b, c, z ∈ C.
Definition 1.23 The hypergeometric supercosine via Gauss superhyperbolic series
is defined as [39]
2Supercos1 (a, b; c; z) =
∞
n=0
(a)2n (b)2n
(c)2n
(−1)n
z2n
(2n)!
, (1.147)
where a, b, c, z ∈ C.
Definition 1.24 The hypergeometricsupertangent via Gauss superhyperbolicseries
is defined as [39]
2Supertan1 (a, b; c; z) =
2Supersin1 (a, b; c; z)
2Supercos1 (a, b; c; z)
, (1.148)
where a, b, c, z ∈ C.
Definition 1.25 The hypergeometric supercotangent via Gauss superhyperbolic
series is defined as [39]
2Supercot1 (a, b; c; z) =
2Supercos1 (a, b; c; z)
2Supersin1 (a, b; c; z)
, (1.149)
where a, b, c, z ∈ C.
Definition 1.26 The hypergeometric supersecant via Gauss superhyperbolic series
is defined as [39]
2Supersec1 (a, b; c; z) =
1
2Supercos1 (a, b; c; z)
, (1.150)
where a, b, c, z ∈ C.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-47-320.jpg)
![26 1 Preliminaries
Definition 1.27 The hypergeometric supercosecant via Gauss superhyperbolic
series is defined as [39]
2Supercosec1 (a, b; c; z) =
1
2Supersin1 (a, b; c; z)
, (1.151)
where a, b, c, z ∈ C.
The formulae for the hypergeometric supertrigonometric functions via Gauss
superhyperbolic series were discovered by Yang in 2020 [39]. In this case, we
present [39]
2Supersin1 (a, b; c; λz)
= 1
2i (2F1 (a, b; c; iλz) −2 F1 (a, b; c; −iλz))
=
∞
n=0
(a)2n+1(b)2n+1
(c)2n+1
(−1)n(λz)2n+1
(2n+1)! ,
(1.152)
2Supercos1 (a, b; c; λz)
= 1
2 (2F1 (a, b; c; iλz) +2 F1 (a, b; c; −iλz))
=
∞
n=0
(a)2n(b)2n
(c)2n
(−1)n(λz)2n
(2n)! ,
(1.153)
2Supertan1 (a, b; c; λz) =
2Supersin1 (a, b; c; λz)
2Supercos1 (a, b; c; λz)
, (1.154)
2Supercot1 (a, b; c; λz) =
2Supercos1 (a, b; c; λz)
2Supersin1 (a, b; c; λz)
, (1.155)
2Supersec1 (a, b; c; λz) =
1
2Supercos1 (a, b; c; λz)
, (1.156)
and
2Supercosec1 (a, b; c; λz) =
1
2Supersin1 (a, b; c; λz)
, (1.157)
where a, b, c, λ, z ∈ C.
Theorem 1.47 Suppose that a, b, c, z ∈ C, and i =
√
−1, then we have
2F1 (a, b; c; iz) = 2Supercos1 (a, b; c; z) + i2Supersin1 (a, b; c; z) . (1.158)
The result was discovered by Yang in 2020 [39].
More generally, we have the following formula:
Suppose that a, b, c, λ, z ∈ C, and i =
√
−1, then we have [39]
2F1 (a, b; c; iλz) = 2Supercos1 (a, b; c; λz) + i2Supersin1 (a, b; c; λz) .
(1.159)](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-48-320.jpg)
![1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 27
1.2.3 The Hypergeometric Superhyperbolic Functions via
Gauss Hypergeometric Series
We now investigate the hypergeometric superhyperbolic functions via Gauss hyper-
geometric series.
Definition 1.28 The hypergeometric superhyperbolic sine via Gauss hypergeomet-
ric series is defined as [39]
2Supersinh1 (a, b; c; z) =
∞
n=0
(a)2n+1 (b)2n+1
(c)2n+1
z2n+1
(2n + 1)!
, (1.160)
where a, b, c, z ∈ C.
Definition 1.29 The hypergeometric superhyperbolic cosine via Gauss hypergeo-
metric series is defined as [39]
2Supercosh1 (a, b; c; z) =
∞
n=0
(a)2n (b)2n
(c)2n
z2n
(2n)!
, (1.161)
where a, b, c, z ∈ C.
Definition 1.30 The hypergeometric superhyperbolic tangent via Gauss hypergeo-
metric series is defined as [39]
2Supertanh1 (a, b; c; z) =
2Supersinh1 (a, b; c; z)
2Supercosh1 (a, b; c; z)
, (1.162)
where a, b, c, z ∈ C.
Definition 1.31 The hypergeometric superhyperbolic cotangent via Gauss hyper-
geometric series is defined as [39]
2Supercoth1 (a, b; c; z) =
2Supercosh1 (a, b; c; z)
2Supersinh1 (a, b; c; z)
, (1.163)
where a, b, c, z ∈ C.
Definition 1.32 The hypergeometric superhyperbolic secant via Gauss hypergeo-
metric series is defined as [39]
2Supersech1 (a, b; c; z) =
1
2Supercosh1 (a, b; c; z)
, (1.164)
where a, b, c, z ∈ C.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-49-320.jpg)
![28 1 Preliminaries
Definition 1.33 The hypergeometric superhyperbolic cosecant via Gauss hyperge-
ometric series is defined as [39]
2Supercosech1 (a, b; c; z) =
1
2Supersinh1 (a, b; c; z)
, (1.165)
where a, b, c, z ∈ C.
The results were proposed by Yang in 2020 [39].
So, there exist [39]
2Supersinh1 (a, b; c; λz)
= 1
2 (2F1 (a, b; c; λz) −2 F1 (a, b; c; −λz))
=
∞
n=0
(a)2n+1(b)2n+1
(c)2n+1
(λz)2n+1
(2n+1)! ,
(1.166)
2Supercosh1 (a, b; c; λz)
= 1
2 (2F1 (a, b; c; λz) +2 F1 (a, b; c; −λz))
=
∞
n=0
(a)2n(b)2n
(c)2n
(λz)2n
(2n)! ,
(1.167)
2Supertanh1 (a, b; c; λz) =
2Supersinh1 (a, b; c; λz)
2Supercosh1 (a, b; c; λz)
, (1.168)
2Supercoth1 (a, b; c; λz) =
2Supercosh1 (a, b; c; λz)
2Supersinh1 (a, b; c; λz)
, (1.169)
2Supersech1 (a, b; c; λz) =
1
2Supercosh1 (a, b; c; λz)
, (1.170)
2Supercosech1 (a, b; c; λz) =
1
2Supersinh1 (a, b; c; λz)
, (1.171)
where a, b, c, λ, z ∈ C.
Theorem 1.48 Suppose that a, b, c, z ∈ C, then we have [39]
2F1 (a, b; c; z) = 2Supercosh1 (a, b; c; z)+2Supersinh1 (a, b; c; z) . (1.172)
So, we have [39]
2F1 (a, b; c; λz) = 2Supercosh1 (a, b; c; λz) + 2Supersinh1 (a, b; c; λz) ,
(1.173)
where a, b, c, λ, z ∈ C.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-50-320.jpg)
![1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 29
Theorem 1.49 Let a, b, c ∈ C. Then we have [39]
2Supersinh1 (a, b; c; 0) = 0. (1.174)
Theorem 1.50 Let a, b, c ∈ C. Then we have [39]
2Supercosh1 (a, b; c; 0) = 1. (1.175)
Theorem 1.51 Let a, b, c ∈ C. Then we have [39]
2Supertanh1 (a, b; c; 0) = 0. (1.176)
Theorem 1.52 Let a, b, c ∈ C. Then we have [39]
2Supersec1 (a, b; c; 0) = 1. (1.177)
The results were discovered by Yang in 2020 [39].
Theorem 1.53 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0,
Re (c1) 0, s ∈ N, and |z| 1, then we have [39]
z
0
tα−1 (z − t)β−1
2F1 a1, a2; c1; λ (z − t)s dt
= B (α, β) zα+β−1 ×2+s F1+s
a1, a2, β
s , β+1
s , · · · , β+s−1
s ;
c1, α+β
s , α+β+1
s , · · · , α+β+s−1
s ;
λzs ,
(1.178)
where λ is a constant.
Theorem 1.54 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (c1)
0,Re (c2) 0, s ∈ N, and |z| 1, then we have [39]
z
0
tα−1 (z − t)β−1
1F2 a1; c1, c2; λ (z − t)s dt
= B (α, β) zα+β−1
1+sF2+s
a1, β
s , β+1
s , · · · , β+s−1
s ;
c1, c2, α+β
s , α+β+1
s , · · · , α+β+s−1
s ;
λzs ,
(1.179)
where λ is a constant.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-51-320.jpg)
![30 1 Preliminaries
Theorem 1.55 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0,
Re (c1) 0, and |z| 1, then we have [39]
z
0
tα−1 (z − t)β−1
2F1 (a1, a2; c1; λ (z − t)) dt
= B (α, β) zα+β−1
3F2
a1, a2, β;
c1, α + β;
λz
,
(1.180)
where λ is a constant and α, β, a1, a2, c1 ∈ C.
Theorem 1.56 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0,
Re (c1) 0, and s ∈ N, then we have [39]
1
−1
(t + 1)α−1
(1 − t)β−1
2F1 a1, a2; c1; λ (1 − t)s dt
= B (α, β) 2α+β−1
2+sF1+s
a1, a2, β
s , β+1
s , · · · , β+s−1
s ;
c1, α+β
s , α+β+1
s , · · · , α+β+s−1
s ;
λ2s ,
(1.181)
where λ is a constant and α, β, a1, a2, c1 ∈ C.
For more details of the applications of the above results, see [39, 62–67].
1.3 Clausen Hypergeometric Series and Supertrigonometric
and Superhyperbolic Functions
In this section, we introduce the definition, theorems, and properties for the Clausen
hypergeometric series.
1.3.1 The Clausen Hypergeometric Series
We now consider the definition and theorems for the Clausen hypergeometric series.
Definition 1.34 (Clausen [68])
The Clausen hypergeometric series of form is defined as [68]
pFq ((a) , (c) ; z)
=p Fq a1, · · · , ap; c1, · · · , cq; z
=p Fq
a1, · · · , ap
c1, · · · , cq
; z
=
∞
n=0
(a1)n···(ap)n
(c1)n···(cq)n
zn
n! ,
(1.182)
where an, cn, z ∈ C and n, p, q ∈ N0.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-52-320.jpg)
![1.3 Clausen Hypergeometric Series and Supertrigonometric and. . . 31
The result, as an extended version of the Gauss hypergeometric series, was
discovered by Clausen in 1828 [68].
Theorem 1.57 (Convergences for the Clausen Hypergeometric Series)
The cases of the convergences of the Clausen hypergeometric series (1.182) hold
for an ∈ CZ−
0 :
(1) if p q, then the series converges absolutely for z ∈ C;
(2) if p = q + 1, then the series converges absolutely for |z| 1 and diverges for
|z| 1, and for |z| = 1 it converges absolutely for Re
q
k=1
ck −
p
k=1
ak
0;
(3) if p q + 1, then the series converges only for z = 0.
The result was reported by Bailey [56], Srivastava and Kashyap [69], Slater [36],
Andrews et al. [26], and Rainville [37].
Theorem 1.58 (The Differential Equation for the Clausen Hypergeometric
Series)
If an, cn, z ∈ C, n, p, q ∈ N0, and |z| 1, the Clausen hypergeometric series is
a solution of the differential equation [69]
(Q (q, μ, cn) ϕ) (z) − (P (p, μ, an) ϕ) (z) = 0, (1.183)
where
(Q (q, μ, cn) ϕ) (z)
=
z d
dz
q
n=1
z d
dz
ϕ (z) + (cn − 1) ϕ (z)
= z d
dz
q
n=1
z d
dz + (cn − 1)
ϕ
(z)
(1.184)
and
(P (p, μ, an) ϕ) (z)
= z
p
n=1
zdϕ(z)
dz + anϕ (z)
= z
p
n=1
z d
dz + an
ϕ
(z).
(1.185)
The result was reported by Srivastava and Kashyap [69], Rainville [37], Andrews et
al. [26], and Luke [14].](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-53-320.jpg)
![32 1 Preliminaries
Theorem 1.59 (The Derivative of the Clausen Hypergeometric Series)
If where an, cn, λ, z ∈ C, n, p, q ∈ N0, Re
q
k=1
ck −
p
k=1
ak
0, and |z| 1,
then there is the derivative of the Clausen hypergeometric series as follows [69]:
d
dz pFq a1, · · · , ap; c1, · · · , cq; z
=
p
n=1
an
q
n=1
cn
pFq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; z
(1.186)
and
1
λ
d
dz pFq a1, · · · , ap; c1, · · · , cq; λz
=
p
n=1
an
q
n=1
cn
pFq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; λz .
(1.187)
The result was discussed by Srivastava and Kashyap [69], Rainville [37], Andrews
et al. [26], and Luke [14].
Theorem 1.60 (Rainville [37])
If p ≤ q + 1, Re (a1) 0,· · · , Re ap 0, Re (c1) 0, · · · , Re cq 0 ,
Re
q
k=1
ck −
p
k=1
ak
0and |z| 1, then there is [37]
pFq
a1, · · · , ap;
c1, · · · , cq;
λz
= Γ (c1)
Γ (a1)Γ (c1−a1)
1
0
ta1−1 (1 − t)c1−a1−1
p−1Fq−1 a2, · · · , ap; c2, · · · , cq; λzt dt,
(1.188)
where λ is a constant.
The result was discovered by Rainville in 1960 (see [37], p.85).
Theorem 1.61 (Askey [70], p.19)
If p ≤ q + 1, Re (a1) 0,· · · , Re ap 0, Re (c1) 0, · · · , Re cq 0 ,
Re
q
k=1
ck −
p
k=1
ak
0and |z| 1, then there is [70]
p+1Fq+1
a1, · · · , ap, a;
c1, · · · , cq, c;
λz
= Γ (c)
Γ (a)Γ (c−a)
1
0
ta−1 (1 − t)c−a−1
pFq a1, a2, · · · , ap; c1, c2, · · · , cq; λzt dt,
(1.189)
where λ is a constant.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-54-320.jpg)
![day, Mr. Calhoun being very active in bringing him back. Mr.
Calhoun was not aware that he was recalling one who was to
prove a successful rival. Mr. Calhoun has not been president; Mr.
Van Buren is so; but the successful rival has a mortal dread of the
great nullifier; a dread so obvious, and causing such a prostration
of all principle and all dignity, as to oblige observers to conclude
that there is more in the matter than they see; that it will come
out some day why the disappointed aspirant is still to be
propitiated, when he seems to be deprived of power to do
mischief. In Society in America I have given an account of the
nullification struggle, and of the irritation, the mysterious
discontent which it has left behind.
[2]
Perhaps Mr. Van Buren may entertain the opinion which many
hold, that that business is not over yet, and that the slavery
question is made a pretext by the nullifiers of the South for a line
of action to which they are impelled by the disappointed personal
ambition of one or two, and the wounded pride of the many, who
cannot endure the contrast between the increase of the free states
of the North and the deterioration of the slave states of the South.
However this may be, to propitiate Mr. Calhoun seems to have
been Mr. Van Buren's great object for a long time past; an object
probably hopeless in itself, and in the pursuit of which he is likely
to lose the confidence of the North far faster than he could, at
best, disarm the enmity of the South.
In the spring of 1836, when Mr. Van Buren was still vice-president,
and the presidential election was drawing near, Mr. Calhoun
brought forward in the Senate his bill (commonly called the Gag
Bill) to violate the postoffice function, by authorizing postmasters
to investigate the contents of the mails, and to keep back all
papers whatsoever relating to the subject of slavery. The bill was,
by consent, read the first and second times without debate; and
the Senate was to be divided on the question whether it should go
to a third reading. The votes were equal, 18 to 18. Where's the](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-75-320.jpg)
![river; and another remarked that they partook with much relish of
the strong wine-sauce served with our puddings.
In addition to other discomforts, we passed the fine scenery of
Little Falls in the night. I was not aware what we had missed till I
traversed the Mohawk valley by a better conveyance nearly two
years afterward. I have described this valley in my other work on
America,
[3]
and must therefore restrain my pen from dwelling on
its beauties here.
The appearance of the berths in the ladies' cabin was so repulsive,
that we were seriously contemplating sitting out all night, when it
began to rain so as to leave us no choice. I was out early in the
misty morning, however, and was presently joined by the rest of
my party, all looking eagerly for signs of Utica being near.
By eight o'clock we were at the wharf. We thought Utica the most
extempore place we had yet seen. The right-up shops, the daubed
houses, the streets running into the woods, all seemed to betoken
that the place had sprung up out of some sudden need. How
much more ancient and respectable did it seem after my return
from the West, where I had seen towns so much newer still! We
were civilly received and accommodated at Bagg's hotel, where we
knew how to value cold water, spacious rooms, and retirement,
after the annoyances of the boat.
Our baggage-master was fortunate in securing a neat, clean stage
to take us to Trenton Falls (14 miles), where we promised each
other to spend the whole day, on condition of being off by five the
next morning, in order to accomplish the distance to Syracuse in
the course of the day. The reason for our economy of time was
not merely that it was late in the season, and every day which
kept us from the Falls of Niagara, therefore, of consequence, but
that our German friend, Mr. O., was obliged to be back in New-
York by a certain day. We all considered a little extra haste and
fatigue a small tax to pay for the privilege of his companionship.](https://image.slidesharecdn.com/18785225-250609142752-a574a10c/85/Theory-And-Applications-Of-Special-Functions-For-Scientists-And-Engineers-Xiaojun-Yang-79-320.jpg)
Theory And Applications Of Special Functions For Scientists And Engineers Xiaojun Yang
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- 5. Xiao-JunYang Theory and Applications of Special Functions for Scientists and Engineers
- 6. Theory and Applications of Special Functions for Scientists and Engineers
- 7. Xiao-Jun Yang Theory and Applications of Special Functions for Scientists and Engineers
- 8. Xiao-Jun Yang School of Mathematics and State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and Technology Xuzhou, Jiangsu, China ISBN 978-981-33-6333-5 ISBN 978-981-33-6334-2 (eBook) https://doi.org/10.1007/978-981-33-6334-2 Mathematics Subject Classification: 33C05, 33C20, 33E12, 44A20, 44A05 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 9. To my family, parents, brother, sister, wife, and my daughters
- 10. Preface The main target of this monograph is to provide the detailed investigations to the newly established special functions involving the Mittag-Leffler, Wiman, Prab- hakar, Miller–Ross, Rabotnov, Lorenzo–Hartley, Sonine, Wright, and Kohlrausch– Williams–Watts functions, Gauss hypergeometric series, and Clausen hypergeomet- ric series. The integral transform operators based on the theory of the Wright and Kohlrausch–Williams–Watts functions may be used to solve the complex problems with power-law behaviors in the light of nature complexity. The topics are important and interesting for scientists and engineers to represent the complex phenomena arising in mathematical physics, engineering, and other applied sciences. The monograph is divided into seven chapters, which are discussed as follows. Chapter 1 introduces the special functions such as Euler gamma function, Pochhammer symbols, Euler beta function, extended Euler gamma function, extended Euler beta function, Gauss hypergeometric series, and Clausen hypergeometric series as well as calculus operators with respect to monotone function containing the power-law calculus, scaling-law calculus, and complex topology calculus as well as calculus operators with respect to logarithmic and exponential functions. Chapter 2 investigates the Wright function, Wright’s generalized hypergeometric function, supertrigonometric and superhyperbolic functions via Wright function, and Wright’s generalized hypergeometric function. The integral representations for the supertrigonometric and superhyperbolic functions are addressed in detail. Some integral transforms via Dunkl transform based on the calculus with respect to power- law function are proposed. Chapter 3 provides the theory of the Mittag-Leffler function, supertrigono- metric functions, and superhyperbolic functions. The integral representations for the Mittag-Leffler function and related functions are addressed, and the general fractional calculus operators are also discussed in detail. The truncated Mittag- Leffler, supertrigonometric,and superhyperbolicfunctions are considered, and some mathematical models are considered to explain the power-law behaviors in material science. vii
- 11. viii Preface Chapter 4 shows the theory of the Wiman function, supertrigonometric functions, and superhyperbolic functions. The integral representations for the Wiman function and related functions are addressed, and the general fractional calculus operators are also discussed in detail. The truncated Wiman, supertrigonometric, and superhyper- bolic functions are considered, and the integral equations as well as mathematical models related to Wiman function are also presented in detail. Chapter 5 addresses the theory of Prabhakar function and proposes the super- trigonometric and superhyperbolic functions via Prabhakar function. The Laplace transforms for the new special functions and integral representations for the super- trigonometric and superhyperbolic functions are discussed in detail. The truncated Prabhakar, supertrigonometric, and superhyperbolic functions are proposed, and the general fractional calculus involving the Prabhakar function is considered. The integral equations and mathematical models related to Prabhakar function are also presented. Chapter 6 presents the Sonine functions, Rabotnov fractional exponential func- tion, Miller–Ross function, and Lorenzo–Hartley functions. The Laplace and Mellin transforms of them are given, and the integral representations for the supertrigono- metric and superhyperbolic functions are also presented in detail. The formulas related to the Mittag-Leffler functions and Wright hypergeometricfunctions are also considered. Chapter 7 illustrates the Kohlrausch–Williams–Watts function and integral repre- sentations. The subtrigonometric functions, subhyperbolic functions, supertrigono- metric functions, and superhyperbolic functions are discussed in detail. Moreover, the Fourier-type series, Fourier-type integral transforms, Laplace-type integral transforms, and Mellin-type integral transforms are also proposed. Xuzhou, China Xiao-Jun Yang August 20, 2020
- 12. Acknowledgments I am particularly indebted to Professor George E. Andrews, Professor H. M. Srivastava, Professor Song Jiang, Professor Bo-Ming Yu, Professor Wolfgang Sprößig, Professor Jeffrey Hoffstein, Professor Nouzha El Yacoubi, Professor Mourad Ismail, Professor Tom H. Koornwinder, Professor Michel L. Lapidus, Professor Norbert Hounkonnou,Professor Simeon Oka, Professor Roland W. Lewis, Professor Manuel López Pellicer, Professor Michael Reissig, Professor George A. Anastassiou, Professor Jin-De Cao, Professor Alain Miranville, Professor Minvydas Ragulskis, Professor Miguel A. F. Sanjuan, Professor Tasawar Hayat, Professor Mahmoud Abdel-Aty, Professor Stefano Galatolo, Professor Dumitru Mihalache, Professor Martin Bohner, Professor Thiab Taha, Professor Salvatore Capozziello, Professor André Keller, Professor Martin Ostoja-Starzewski, Professor Vukman Bakic, Professor J. A. Tenreiro Machado, Professor Dumitru Baleanu, Professor Sohail Nadeem, Professor Mauro Bologna, Professor Hui-Ming Wang, Professor Long Jiang, Professor Sheng-Bo Li, Professor Carlo Cattani, Professor Semyon Yakubovich, Professor Ayman S. Abdel-Khalik, Professor Irene Maria Sabadini, Professor Mario Di Paola, Professor Mokhtar Kirane, Professor Giuseppe Failla, Professor Reza Saadati, Professor Yusif Gasimov, and Professor Ivanka Stamova. My special thanks go to Professor He-Ping Xie, Professor Feng Gao, Professor Fu-Bao Zhou, Professor Hong-Wen Jing, Professor Xiao-Zhao Li, Professor Yang Ju, Professor Hong-Wei Zhou, and Professor Ming-Zhong Gao, and the financial support of the Yue-Qi Scholar of the China University of Mining and Technology (grant no. 04180004). It is my pleasure to thank my students who support me to handle the Latex files. Finally, I also wish to express my special thanks to Springer staff, especially, Daniel Wang and Dimana Tzvetkova, for their cooperation in the production process of this book. ix
- 13. Contents 1 Preliminaries ................................................................. 1 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function, and Related Functions ................................ 1 1.1.1 The Euler Gamma Function ................................. 1 1.1.2 The Pochhammer Symbols and Related Formulas ......... 9 1.1.3 The Euler Beta Function ..................................... 12 1.1.4 The Extended Euler Gamma Functions ..................... 15 1.1.5 The Extended Euler Beta Functions......................... 18 1.2 Gauss Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions ............................................ 22 1.2.1 The Gauss Hypergeometric Series .......................... 22 1.2.2 The Hypergeometric Supertrigonometric Functions via Gauss Superhyperbolic Series ............... 25 1.2.3 The Hypergeometric Superhyperbolic Functions via Gauss Hypergeometric Series ........................... 27 1.3 Clausen Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions ....................................... 30 1.3.1 The Clausen Hypergeometric Series ........................ 30 1.3.2 The Hypergeometric Supertrigonometric Functions via Clausen Superhyperbolic Series ............. 37 1.3.3 The Hypergeometric Superhyperbolic Functions via Clausen Superhyperbolic Series......................... 38 1.3.4 The Series Representations for the Special Functions ..... 40 1.4 The Laplace and Mellin Transforms .................................. 69 1.4.1 The Laplace Transforms for the Special Functions ........ 69 1.4.2 The Mellin Transforms for the Special Functions .......... 71 1.5 Calculus with Respect to Monotone Functions ...................... 74 1.5.1 The Newton–Leibniz Calculus .............................. 74 1.5.2 Calculus with Respect to Monotone Function .............. 75 1.5.3 The Special Integral Equations .............................. 86 xi
- 14. xii Contents 1.5.4 Generalized Functions and Anomalous Linear Viscoelasticity via Derivative with Respect to Another Function ............................................ 97 1.6 Derivative and Integral with Respect to Power-Law Function ...... 99 1.6.1 The Derivative with Respect to Power-Law Function...... 99 1.6.2 The Integral with Respect to Power-Law Function......... 101 1.6.3 The Scaling-Law Calculus................................... 108 1.6.4 The Special Formulas via Scaling-Law Calculus........... 120 1.6.5 Other Calculus Operators with Respect to Monotone Functions ......................................... 138 2 Wright Function and Integral Transforms via Dunkl Transform ...... 147 2.1 The Special Functions Related to Wright Function and Integral Representations ............................................... 147 2.1.1 The Wright’s Generalized Hypergeometric Function ...... 147 2.1.2 The Integral Representations via Wright’s Generalized Hypergeometric Function ..................... 150 2.1.3 The Integral Transforms for the Generalized Wright Functions............................................. 154 2.1.4 The Supertrigonometric Functions via Wright’s Generalized Hypergeometric Function ..................... 157 2.1.5 The Superhyperbolic Functions via Wright’s Generalized Hypergeometric Function ..................... 168 2.1.6 The Supertrigonometric Functions via Wright Function ... 183 2.1.7 The Superhyperbolic Functions via Wright Function ...... 189 2.2 The Truncated Wright’s Generalized Hypergeometric Function .... 204 2.3 The Integral Transforms via Dunkl Transform ...................... 206 2.3.1 The Dunkl Transform ........................................ 206 2.3.2 New Integral Transforms of First Type ..................... 221 2.3.3 New Integral Transforms of Second Type .................. 229 2.3.4 New Integral Transforms of Third Type .................... 236 3 Mittag-Leffler, Supertrigonometric, and Superhyperbolic Functions...................................................................... 255 3.1 The Mittag-Leffler Function: History, Definitions, Properties, and Theorems ............................................. 255 3.1.1 The Mittag-Leffler Function ................................. 255 3.1.2 Special Integral Representations ............................ 259 3.1.3 The Integral Transforms for the Mittag-Leffler Functions ..................................................... 272 3.1.4 The Supertrigonometric Functions via Mittag-Leffler Function ...................................... 274 3.1.5 The Superhyperbolic Functions via Mittag-Leffler Function ...................................................... 279 3.1.6 The Pre-supertrigonometric Functions via Mittag-Leffler Function ...................................... 284
- 15. Contents xiii 3.1.7 The Pre-superhyperbolic Functions via Mittag-Leffler Function ...................................... 292 3.1.8 The Laplace Transforms of the Special Functions via Mittag-Leffler Function.................................. 301 3.2 Analytic Number Theory Involving the Mittag-Leffler Function ... 302 3.2.1 The Basic Formulas Involving the Mittag-Leffler Function ...................................................... 302 3.2.2 The Generalized Hyperbolic Function ...................... 307 3.3 The Special Integral Equations via Mittag-Leffler Function and Related Functions ................................................. 307 3.3.1 The Integral Equations of Volterra Type .................... 307 3.3.2 The Integral Equations of Fredholm Type .................. 311 3.4 The Integral Representations for the Special Function via Mittag-Leffler Function ............................................... 315 3.5 The Fractional Equations via Mittag-Leffler Function and Related Functions...................................................... 317 3.6 General Fractional Calculus Operators with Mittag-Leffler Function ................................................................ 320 3.6.1 Hille–Tamarkin General Fractional Derivative ............. 320 3.6.2 Hille–Tamarkin General Fractional Integrals ............... 323 3.6.3 Liouville–Weyl–Hille–Tamarkin Type General Fractional Calculus .......................................... 324 3.6.4 Hilfer–Hille–Tamarkin Type General Fractional Derivative with Nonsingular Kernel ........................ 326 3.6.5 Hille–Tamarkin General Fractional Derivative with Respect to Another Function........................... 328 3.6.6 Hille–Tamarkin General Fractional Integrals with Respect to Another Function ................................ 330 3.6.7 Liouville–Weyl–Hille–Tamarkin Type General Fractional Calculus with Respect to Another Function .... 331 3.6.8 Hilfer–Hille–Tamarkin Type General Fractional Derivative with Respect to Another Function .............. 334 3.7 The Integral Representations Related to Mittag-Leffler Function ... 336 3.8 The Relationship Between Mittag-Leffler Function and Wright’s Generalized Hypergeometric Function ................ 338 3.9 The Truncated Mittag-Leffler Functions and Related Functions .... 355 3.10 Applications in Anomalous Linear Viscoelasticity .................. 365 4 Wiman, Supertrigonometric, and Superhyperbolic Functions ......... 367 4.1 The Wiman Function: History, Definitions, Properties, and Theorems............................................................... 367 4.1.1 The Wiman Function ........................................ 367 4.1.2 The Supertrigonometric Functions via Wiman Function... 377 4.1.3 The Superhyperbolic Functions via Wiman Function ...... 383
- 16. xiv Contents 4.1.4 The Pre-supertrigonometric Functions via Wiman Function ...................................................... 391 4.1.5 The Pre-superhyperbolic Functions via Wiman Function ...................................................... 395 4.1.6 Some Special Cases via Wiman Function .................. 399 4.1.7 The Special Integral Equations via Viman Function and Related Functions............................. 424 4.1.8 The Integral Representations Related to Viman Function ...................................................... 430 4.1.9 The Special Cases Based on the Wiman Function ......... 432 4.2 The Integral Representations Related to Wiman, Supertrigonometric, and Superhyperbolic Functions ................ 439 4.3 The Truncated Wiman Functions and Related Functions ............ 470 4.4 General Fractional Derivatives Within the Wiman Kernel........... 487 4.4.1 General Fractional Derivatives Within the Wiman Kernel......................................................... 487 4.4.2 Hilfer-Type General Fractional Derivatives with the Wiman Kernel ............................................ 489 4.4.3 General Fractional Derivatives with Respect to Another Function via Wiman Function ..................... 491 4.5 Applications............................................................ 496 5 Prabhakar, Supertrigonometric, and Superhyperbolic Functions ..... 499 5.1 The Prabhakar Function: History, Definitions, Properties, and Theorems .......................................................... 499 5.1.1 The Prabhakar Function ..................................... 499 5.1.2 The Supertrigonometric Functions via Prabhakar Function ...................................................... 523 5.1.3 The Superhyperbolic Functions via Prabhakar Function... 528 5.1.4 The Pre-Supertrigonometric Functions via Prabhakar Function .......................................... 533 5.1.5 The Pre-Superhyperbolic Functions via Prabhakar Function ...................................................... 537 5.1.6 The Pre-Supertrigonometric Functions with Power Law via Prabhakar Function ......................... 542 5.1.7 The Pre-Superhyperbolic Functions with Power Law via Prabhakar Function................................. 546 5.1.8 The Pre-Supertrigonometric Functions with the Parameter via Prabhakar Function .......................... 551 5.1.9 The Pre-Superhyperbolic Functions with the Parameter via Prabhakar Function .......................... 555 5.1.10 The Pre-Supertrigonometric Functions with the Power Law and Parameter via Prabhakar Function ........ 560 5.1.11 The Pre-Superhyperbolic Functions with the Power Law and Parameter via Prabhakar Function ........ 564
- 17. Contents xv 5.2 The Integral Representations for Special Functions Related to Prabhakar Function ................................................. 568 5.3 The Truncated Prabhakar Functions and Related Functions......... 594 5.3.1 The Truncated Prabhakar Functions ........................ 594 5.3.2 Other Special Functions Related to Prabhakar Function ... 609 5.4 General Fractional Calculus Operators via Prabhakar Function..... 617 5.4.1 Kilbas–Saigo–Saxena Derivative via Prabhakar Function ...................................................... 617 5.4.2 Garra–Gorenflo–Polito–Tomovski Derivative via Prabhakar Function .......................................... 618 5.4.3 Prabhakar-Type Integrals .................................... 619 5.4.4 Kilbas–Saigo–Saxena-Type Derivative via Prabhakar Function .......................................... 620 5.4.5 Garra–Gorenflo–Polito–Tomovski-Type Derivative via Prabhakar Function .......................... 621 5.4.6 Prabhakar-Type Integrals .................................... 622 5.4.7 Hilfer-Type Derivative via Prabhakar Function ............ 623 5.4.8 Kilbas–Saigo–Saxena-Type Derivative with Respect to Another Function ................................ 625 5.4.9 Garra–Gorenflo–Polito–Tomovski-Type Derivative with Respect to Another Function .............. 626 5.4.10 Prabhakar-Type Integrals with Respect to Another Function ...................................................... 626 5.4.11 Kilbas–Saigo–Saxena-Type Derivative with Respect to Another Function ................................ 627 5.4.12 Garra–Gorenflo–Polito–Tomovski-Type Derivative with Respect to Another Function .............. 628 5.4.13 Prabhakar-Type Integrals with Respect to Another Function ...................................................... 629 5.4.14 Hilfer-Type Derivative with Respect to Another Function via Prabhakar Function............................ 630 5.5 Applications............................................................ 632 5.5.1 The Integral Equations in the Kernel of New Special Functions ............................................ 632 5.5.2 Anomalous Viscoelasticity and Diffusion .................. 643 6 Other Special Functions Related to Mittag-Leffler Function ........... 647 6.1 The Sonine Functions: History, Definitions, and Properties ......... 647 6.1.1 The Sonine Functions of First Type ......................... 647 6.1.2 The Supertrigonometric Functions via Sonine Function of First Type ....................................... 660 6.1.3 The Superhyperbolic Functions via Sonine Function of First Type ....................................... 662 6.1.4 The Integral Representations of the Supertrigonometric and Superhyperbolic Functions ....... 665
- 18. xvi Contents 6.1.5 The Sonine Functions of Second Type...................... 669 6.1.6 The Supertrigonometric Functions via Sonine Function of Second Type .................................... 673 6.1.7 The Superhyperbolic Functions via Sonine Function of Second Type .................................... 675 6.1.8 The Sonine Function of Third Type ......................... 678 6.1.9 The Supertrigonometric Functions via Sonine Function of Third Type ...................................... 680 6.1.10 The Superhyperbolic Functions via Sonine Function of Third Type ...................................... 682 6.1.11 The Integral Representations Related to Sonine Function of Third Type ...................................... 685 6.1.12 The Integral Representations for the Sonine Function of Second Type .................................... 689 6.2 The Rabotnov Fractional Exponential Function ..................... 694 6.2.1 The Rabotnov Fractional Exponential Function: History and Properties ....................................... 695 6.2.2 The Supertrigonometric Functions via Rabotnov Function ...................................................... 698 6.2.3 The Superhyperbolic Functions via Rabotnov Function ... 701 6.2.4 The Supertrigonometric Functions via Rabotnov Type Function ................................................ 703 6.2.5 The Superhyperbolic Functions via Rabotnov Type Function ................................................ 705 6.2.6 The Integral Representations of the Supertrigonometric and Superhyperbolic Functions ....... 708 6.2.7 The Gauss–Rabotnov Type Functions ...................... 712 6.3 The Miller–Ross Function............................................. 715 6.3.1 The Miller–Ross Function: History and Properties ........ 715 6.3.2 The Supertrigonometric Functions via Miller–Ross Function ........................................ 718 6.3.3 The Superhyperbolic Functions via Miller–Ross Function ...................................................... 720 6.3.4 The Integral Representations Related to Miller–Ross Function ........................................ 723 6.4 The Lorenzo–Hartley Functions ...................................... 727 6.4.1 The Lorenzo–Hartley Function of First Type............... 728 6.4.2 The Supertrigonometric Functions via Lorenzo–Hartley Function of First Type.................... 730 6.4.3 The Superhyperbolic Functions via Lorenzo–Hartley Function of First Type.................... 733 6.4.4 The Integral Representations for the Special Functions Related to the Lorenzo–Hartley Function of First Type ....................................... 736 6.4.5 The Lorenzo–Hartley Function of Second Type............ 741
- 19. Contents xvii 6.4.6 The Supertrigonometric Functions via Lorenzo–Hartley Function of Second Type................. 743 6.4.7 The Superhyperbolic Functions via Lorenzo–Hartley Function of Second Type................. 746 6.4.8 The Integral Representations for the Special Functions Related to the Lorenzo–Hartley Function of Second Type .................................... 749 7 Kohlrausch–Williams–Watts Function and Related Topics ............. 757 7.1 The Kohlrausch–Williams–Watts Function: History, Definitions, and Properties ............................................ 757 7.1.1 The Kohlrausch–Williams–Watts Function ................. 757 7.1.2 The Subtrigonometric Functions via Kohlrausch–Williams–Watts Function ...................... 764 7.1.3 The Subhyperbolic Functions via Kohlrausch–Williams–Watts Function ...................... 767 7.1.4 The Subtrigonometric Functions via Kohlrausch–Williams–Watts Type Function................ 769 7.1.5 The Subhyperbolic Functions via Kohlrausch–Williams–Watts Type Function................ 772 7.1.6 The Integral Representations Associated with Kohlrausch–Williams–Watts Function ...................... 774 7.1.7 The Special Functions with Complex Topology............ 787 7.1.8 Subsurfaces and Geometric Representations Related to Kohlrausch–Williams–Watts Function .......... 789 7.2 The Fourier-Type Series Theory via Subtrigonometric Series with Respect to Monotone Function .......................... 797 7.2.1 Theory of Fourier Series: History and Properties .......... 797 7.2.2 The Subtrigonometric and Subhyperbolic Functions with Respect to Monotone Function............. 799 7.2.3 The Subtrigonometric Functions with Respect to Monotone Function .......................................... 800 7.2.4 The Subhyperbolic Functions with Respect to Monotone Function .......................................... 803 7.3 Theory of Subtrigonometric Series with Respect to Monotone Function .................................................... 807 7.3.1 The Subtrigonometric Series with Respect to Monotone Function .......................................... 807 7.3.2 The Subtrigonometric Series with Respect to Scaling-Law Function ....................................... 824 7.3.3 Theory of Subtrigonometric Series with Respect to Complex and Power-Law Functions ..................... 833 7.3.4 Applications .................................................. 841
- 20. xviii Contents 7.4 The Fourier-Like Integral Transforms via Subtrigonometric Functions with Respect to Monotone Function ...................... 844 7.4.1 Fourier Transform: History, Concepts, and Theorems ..... 845 7.4.2 The Integral Transform Operator with Respect to Monotone Function of First Type ........................... 845 7.4.3 The Integral Transform Operator with Respect to Monotone Function of Second Type ........................ 847 7.4.4 The Integral Transform Operator with Respect to Monotone Function of Third Type .......................... 848 7.4.5 The Integral Transform Operator with Respect to Monotone Function of Fourth Type ......................... 849 7.4.6 The Integral Transform Operator with Respect to Monotone Function of Fifth Type ........................... 850 7.4.7 The Integral Transform Operator with Respect to Monotone Function of Sixth Type .......................... 851 7.4.8 The Integral Transform Operator with Respect to Monotone Function of Seventh Type ....................... 852 7.4.9 The Integral Transform Operator with Respect to Monotone Function of Eighth Type ......................... 854 7.4.10 The Integral Transform Operator with Respect to Monotone Function of Ninth Type .......................... 855 7.4.11 The Integral Transform Operator with Respect to Power-Law Function of Second Type....................... 856 7.4.12 The Integral Transform Operator with Respect to Power-Law Function of Fifth Type.......................... 857 7.4.13 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type ..................... 857 7.4.14 The Integral Transform Operator with Respect to Scaling-Law Function of Fifth Type ........................ 858 7.4.15 Applications .................................................. 859 7.5 The Laplace-Like Transforms via Subtrigonometric Functions with Respect to Monotone Function ...................... 862 7.5.1 Laplace Transform: History, Concepts, and Theorems..... 862 7.5.2 The Integral Transform Operator with Respect to Monotone Function of First Type ........................... 863 7.5.3 The Integral Transform Operator with Respect to Monotone Function of Second Type ........................ 864 7.5.4 The Integral Transform Operator with Respect to Monotone Function of Third Type .......................... 866 7.5.5 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type ..................... 867 7.5.6 The Integral Transform Operator with Respect to Power-Law Function of Second Type....................... 868 7.5.7 The Bilateral Integral Transform Operator with Respect to Monotone Function of Fourth Type............. 869
- 21. Contents xix 7.5.8 The Bilateral Integral Transform Operator with Respect to Monotone Function of Second Type ............ 871 7.5.9 The Bilateral Integral Transform Operator with Respect to Monotone Function of Third Type .............. 872 7.5.10 The Bilateral Integral Transform Operator with Respect to Scaling-Law Function of Second Type ......... 873 7.5.11 The Bilateral Integral Transform Operator with Respect to Power-Law Function of Second Type .......... 874 7.5.12 Applications .................................................. 875 7.6 The Mellin-Like Transforms via Subtrigonometric Functions with Respect to Monotone Function ...................... 877 7.6.1 Mellin Transform: History, Concepts, and Theorems ...... 877 7.6.2 The Integral Transform Operator with Respect to Monotone Function of First Type ........................... 877 7.6.3 The Integral Transform Operator with Respect to Monotone Function of Second Type ........................ 879 7.6.4 The Integral Transform Operator with Respect to Scaling-Law Function of Second Type ..................... 881 7.6.5 The Integral Transform Operator with Respect to Power-Law Function of Second Type....................... 882 References......................................................................... 883
- 22. About the Author Xiao-Jun Yang, PhD, is Professor of Applied Math- ematics and Mechanics at China University of Min- ing and Technology, Xuzhou, China. His scientific interests include mathematical physics, fractional calculus and applications, fractals, mechanics, ana- lytic number theory, integral transforms, and spe- cial functions. He is a recipient of the Atanasije Stojkovič Medal, Belgrade, Serbia (2021). Profes- sor Yang was awarded the Abel Award (Istanbul, Turkey, 2020) for his achievements in the area of fractional calculus and its applications. He was also awarded the Obada-Prize, Cairo, Egypt (2019). He is a recipient of the Young Scientist Award (2019) for contributions in developing local fractional cal- culus at ICCMAS-2019, Istanbul, Turkey, and the Springer Distinguished Researcher Award (2019) at ICMMAAC-2019, Jaipur, India. He is a highly cited researcher (2021, 2020, and 2019, Clarivate Analyt- ics) in mathematics and Elsevier Most Cited Chinese Researcher in Mathematics (2017, 2018, 2019, and 2020). Professor Yang is one of the scientific com- mittee members of 10th edition of the Pan African Congress of Mathematicians. He is the author and co-author of seven monographs for Elsevier, Springer Nature, CRC, World Science, and Asian Academic, and co-editor of one edited book for De Gruyter. xxi
- 23. Chapter 1 Preliminaries Abstract In this chapter, we investigate the special functions and operator calculus. At first, the Euler gamma function, Pochhammer symbols, Euler beta function, extended Euler gamma function, and extended Euler beta function are introduced. Then, the Gauss hypergeometric series, Clausen hypergeometric series, super- trigonometric and superhyperbolic functions, and Laplace and Mellin transforms are presented. Finally, the calculus operators with respect to monotone function are discussed and the mathematical models in applied sciences are also reported in detail. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function, and Related Functions In this section, we present the Euler gamma function, Pochhammer symbols, Euler beta function, extended Euler gamma function, and extended Euler beta function. 1.1.1 The Euler Gamma Function In this part, we introduce the Euler gamma function. Let C, R, Z, N be the sets of the complex numbers, real numbers, integrals, and natural numbers, respectively. Let Z+, R+, Z−, and R− be the sets of the positive integrals, positive real numbers, and negative integral numbers, and negative real numbers. Let Z− 0 = Z− ∪ 0 and N0 = N ∪ 0. Let Re (x) denote the real part of x if x ∈ C. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 X.-J. Yang, Theory and Applications of Special Functions for Scientists and Engineers, https://doi.org/10.1007/978-981-33-6334-2_1 1
- 24. 2 1 Preliminaries Definition 1.1 (Euler [1]) The gamma function due to Euler is defined as Γ (z) = ∞ 0 e−t tz−1 dt, (1.1) where Re (z) 0 and z ∈ C. The formula was discovered by Euler in 1729 (see [1], p.1), and the notation Γ (z) was introduced by Legendre in 1814 (see [2], p.476). Theorem 1.1 (Weierstrassian Product [3]) If z ∈ CZ− 0 with Z− 0 =: {0, −1, −2, · · ·} and γ := lim n→∞ n k=1 1 k − log n is the Euler constant, the Gamma function was given as [3] Γ (z) = e−γ z z ∞ k=1 1 + z k −1 e z k . (1.2) Moreover, Γ (z) is analytic except at the points z ∈ Z− 0 , where it has simple poles [4]. The formula for the Weierstrassian product was discovered by Weierstrass in 1856 [3] and by Newman in 1848 [5], respectively, and the proofs were published by Hölder [6], Moore [7], and Baines [8]. Definition 1.2 (Euler [1]) Let Re (z) 0 and z ∈ C. Then the Euler’s functional equation states Γ (z + 1) = zΓ (z) . (1.3) The result is the Euler’s functional equation discovered by Euler in 1729 [9] and reported by Weierstrass [3], Brunel [10], Gronwall [11], and Olver [12]. Theorem 1.2 (Euler [1]) If z ∈ N0, then we have Γ (z + 1) = z!. (1.4) The result is the Euler’s functional equation discovered by Euler in 1729 [1, 2] and discussed by Weierstrass [4], Brunel [10], and Gronwall [11]. Theorem 1.3 (Euler [1]) Γ 1 2 = √ π. (1.5) This work was discovered by Euler in 1729 [11] and discussed in Bell [13], Luke [14], and Bendersky [15].
- 25. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 3 Theorem 1.4 (Euler) If n, j ∈ N, then we have n j=1 Γ 1 − j n = (2π) n−1 2 n 1 2 . (1.6) The result was reviewed by Gronwall in 1916 [11]. Theorem 1.5 (Winckler [16]) If z ∈ C and g, k, j, k, l, m, n ∈ N, then we have n−1 j=0 Γ hz + hj n m−1 l=0 Γ gz + gl m = h g hgz+ hg−h−g 2 (2π) h−g 2 . (1.7) The result was discovered by Winckler in 1856 [16] and reviewed by Gronwall in 1916 [11]. Theorem 1.6 (Schlömilch [17] and Newman [5]) If z ∈ C and k ∈ N, then we have 1 Γ (z) = eγ z z ∞ k=1 1 + z k e− z k . (1.8) The result was discovered by Schlömilch in 1844 [17] and by Newman [18]. Theorem 1.7 (Whittaker [19]) If Re (z) 0, z ∈ C, and k ∈ N, then we have ∞ 0 e−kt tz−1 dt = Γ (z) kz . (1.9) The result was first reported by Whittaker in 1902 (see [19], p.184) and further reported by Whittaker and Watson in 1920 [20]. Theorem 1.8 (Whittaker [19]) If α, β ∈ C, Re (α) 0, and Re (β) 0, then we have π 2 0 cosα−1 t sinβ−1 tdt = 1 2 Γ α 2 Γ α 2 Γ α+β 2 . (1.10) The result was first defined by Whittaker in 1902 (see [19], p.191) and further reported by Whittaker and Watson in 1920 [20].
- 26. 4 1 Preliminaries Theorem 1.9 (Titchmarsh [21]) If α, β ∈ C, Re (α) 0, Re (β) 0, and Re (α + β) 1, then we have ∞ −∞ 1 Γ (α + t) Γ (β − t) dt = 2α+β−1 Γ (α + β − 1) . (1.11) The result was first reported in the Titchmarsh’s monograph [21]. Theorem 1.10 (Titchmarsh [21]) If Re (α) −1, Re (β) −1, and Re (α + β) −1, then we have f ∞ k=1 k (α + β + k) (α + k) (β + k) = Γ (α + 1) Γ (β + 1) Γ (α + β + 1) . (1.12) The result was first presented in the Titchmarsh’s monograph [21]. Theorem 1.11 (Titchmarsh [21]) If z ∈ C and k, n ∈ N, then we have n k=1 1 − z kn = − n k=1 Γ −e2πi k−1 n z 1 n −1 . (1.13) The result was first reported in the Titchmarsh’s monograph [21]. Theorem 1.12 (Euler [22]) Let z ∈ C and Re (z) 0. Then we have the Euler’s completion formula as follows: Γ (z) Γ (1 − z) = π sin (πz) (1.14) and sin (πz) = πz n k=1 1 − z2 k2 . (1.15) The result is the Euler’s completion formula due to Euler [22]. For more details of the results, readers refer to Weierstrass [4], Manocha and Srivastava [23], Luke [14], Bell [13], Godefroy [24] and Tannery [25]. Theorem 1.13 (Legendre [2], p.485) The Legendre duplication formula states Γ (2z) Γ 1 2 = 22z−1 Γ (z) Γ z + 1 2 , (1.16) where z ∈ CZ− 0 .
- 27. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 5 The Legendre’s duplication formula was first discovered by Legendre in 1809 (see [2], p.477). For more details of the Legendre duplication formula, readers refer to Gronwall [11], Andrews et al. [26], and Manocha and Srivastava [23]. Theorem 1.14 (Gauss [27]) If z ∈ C 0, − j m with j m and j, m ∈ N, then we have Γ (mz) = (2π) 1−m 2 mmz− 1 2 m j=1 Γ z + j − 1 m . (1.17) The result is the Gauss’ multiplication formula due to Gauss [27]. For more details of the Gauss’ multiplication formula, readers refer to Winckler [28], Gronwall [11], Manocha and Srivastava [23], and Andrews et al. [26]. Theorem 1.15 (Weierstrass [3]) If z ∈ C/Z− 0 , then we have Γ 1 2 − z Γ 1 2 +z = πsec (πz) = π cos(πz) = 2π eiπz+e−iπz . (1.18) The result was discovered by Weierstrass [4] and reported by Bell in 1968 [13] and by Luke in 1969 [14]. Theorem 1.16 Γ 1 2 + iz Γ 1 2 − iz = π cosh(πz) = 2π eπz+e−πz , (19) Γ 1 2 + iz 2 = 2π eπz + e−πz (1.19) and Γ (iz) Γ (−iz) = π −iz sin (πzi) = 2π z eπz − e−πz , (1.20) where |z| → ∞. The results were reported by different researchers, for example, Lerch [29], Godefroy [30], Stieltjes [31], Bateman [32], and Andrews et al. [26]. Theorem 1.17 n−1 j=1 Γ j n Γ 1 − j n = (2π)n−1 n , (1.21)
- 28. 6 1 Preliminaries Γ −n + 1 2 = (−1)n 2n√ π (2n − 1)! , (1.22) Γ n + 1 2 = (2n − 1)! √ π 2n , (1.23) Γ (n + z) Γ (n − z) = πz sin (πz) ((n − 1)!)2 n−1 j=1 Γ 1 − z2 j (1.24) and Γ n + 1 2 + z Γ n + 1 2 − z = Γ n + 1 2 2 cos (πz) n j=1 Γ 1 − 4z2 (2j − 1)2 , (1.25) where n ∈ N and z ∈ CZ− 0 . The results were reported by Weierstrass in 1856 [4]. Let us introduce the Temme function which is related to the ratio of two gamma functions [35]. Definition 1.3 The Temme function is defined as Γ ∗ (z) = Γ (z) √ 2πzz− 1 2 e−z , (1.26) where z ∈ C and Re (z) 0. The result was defined by in Temme’s book (see [35], p.66). Theorem 1.18 Let z, a, b ∈ C, Re (z) 0, Re (z + a) 0, and Re (z + b) 0. Then the ratio of two gamma functions is shown as follows: Γ (z + a) Γ (z + b) = zα−b Γ ∗ (z + a) Γ ∗ (z + b) Q (z, a, b) , (1.27) where Q (z, a, b) = 1 + a z a− 1 2 1 + b z 1 2 −b e z ln 1+ a z − a z −ln 1+ b z + b z . (1.28) The result was discovered by Temme in the book (see [35], p.67).
- 29. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 7 In an alternative manner, we have (see [35], p.67) Γ (z + a) Γ (z + b) = 1 Γ (b − a) 1 0 tz+a−1 (1 − t)b−a−1 dt, (1.29) where a, b, z ∈ C and Re (b − a) 0. The result was reported by Temme in the book (see [35], p.67). Here, we introduce the interested formula reported in the book (see [35], p.72) as follows: Theorem 1.19 Let a, b, z ∈ C , Re (a) 0, Re (b) 0, and Re (z) 0. Then we have ∞ 0 tz−1 e−atb dt = 1 b Γ z b a− z b . (1.30) The result was reported by Temme (see [35], p.72). There are some special cases of (1.29) as follows: ∞ 0 tz−1 e−at dt = a−z Γ (z) , (1.31) ∞ 0 e−tb dt = Γ 1 b + 1 , (1.32) ∞ 0 e−atb dt = 1 b Γ z b a− z b , (1.33) ∞ 0 e−at2 dt = 1 2 Γ z 2 a− z 2 , (1.34) ∞ 0 tz−1 e−at2 dt = 1 2 Γ z 2 a− z 2 , (1.35) ∞ 0 tz−1 e−t2 dt = 1 2 Γ z 2 (1.36)
- 30. 8 1 Preliminaries and ∞ 0 tz−1 e−tb dt = 1 b Γ z b , (1.37) where a, b, z ∈ C , Re (a) 0, Re (b) 0, and Re (z) 0. There are useful formulas as follows [35]: Γ n+ 1 2 = (2n)! √ π 22nn! , (1.38) Γ −n+ 1 2 = (−1)n √ π22n n! (2n)! , (1.39) Γ 1 2 − z Γ z+ 1 2 = π cos (πz) , z − 1 2 / ∈ Z, (1.40) Γ 1 2 − xi Γ xi+ 1 2 = π cosh (πx) , (1.41) ∞ 0 tz−1 sin tdt = Γ (z) sin πz 2 (1.42) and ∞ 0 tz−1 cos tdt = Γ (z) cos πz 2 , (1.43) where z ∈ C , Re (z) 0, n ∈ N and x ≥ 0. Making use of (1.30) and taking a = i = √ −1, we have ∞ 0 tz−1 e−it dt = i−z Γ (z) , (1.44) ∞ 0 tz−1e−itdt = ∞ 0 tz−1 (cos t − i sin t) dt = ∞ 0 tz−1 cos tdt − i ∞ 0 tz−1 sin tdt (1.45)
- 31. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 9 and i−zΓ (z) = e− zπi 2 Γ (z) = cos zπ 2 − sin zπ 2 Γ (z) , (1.46) where (iθ)z = |θ| e− zπisgn(z) 2 , z ∈ C and Re (z) 0, such that we obtain (1.42) and (1.43). 1.1.2 The Pochhammer Symbols and Related Formulas We now introduce the Pochhammer symbols and related theorems. Definition 1.4 (Pochhammer [34]) The Pochhammer symbol is defined as [34] (α)k = k n=1 (α + n − 1) = Γ (α+k) Γ (α) = 1 (k = 0) α (α + 1) · · · (α + k − 1) (k ∈ N0) (1.47) and (α)0 = 1, (1.48) where α ∈ C and k, n ∈ N. The Pochhammer symbol was first suggested by Pochhammer in 1870 [34]. The notation was first used by Pochhammer in 1870 [34] and Weierstrass noticed in 1856 that [4] Γ (α + k) = α (α + 1) · · · (α + k − 1) Γ (α) (k ∈ N0) . (1.49) For more information, readers may refer to the monograph [33]. Moreover, there is (see[4, 11, 36]) lim k→∞ (α)k = 1 Γ (α) , (1.50) where α ∈ CC− 0 and k ∈ N.
- 32. 10 1 Preliminaries Suppose that α = −n and n ∈ N0, then there is (see [36], p.3) (α)k = (−n)k , n ≥ k −, n k. (1.51) Theorem 1.20 (Euler [1]) If z ∈ CZ− 0 , then we have Γ (z) = lim n→∞ nz (z)n+1 . (1.52) The result was discovered by Euler in 1729 [9], reported by Weierstrass in 1856 [3], and discussed by Gronwall in 1916 [11]. Theorem 1.21 There exist (α)k (α + k)n = (α)n+k , (1.53) and (α + k)m−k = (α)m (α)k , (1.54) where α ∈ CZ− 0 and n, k ∈ N. The first formula of the results was reported by Rainville in 1960 (see [37], p.59) and the second formula was suggested by Slater in 1966 (see [36], p.31). There are some useful formulas as follows: (−z)n = (−1)n (z − n + 1) n, (1.55) (z)2n = 22n z 2 n z 2 + 1 2 n, (1.56) (z)2n+1 = 22n+1 z 2 n+1 z 2 + 1 2 n, (1.57) where z ∈ C and n ∈ N. Theorem 1.22 Let j, k, m, n ∈ N0, k ≤ n and α, β ∈ CZ, then we have (1)n = n!, (1.58)
- 33. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 11 α n = α(α−1)···(α−k−1) n! = Γ (α+1) n!Γ (α−n+1) = 1 n!(α+1)n = (−1)n(−α)n n! , (1.59) Γ (α + 1) Γ (α − n + 1) = (−1)n (−α)n , (1.60) 1 (m − n)! = (−1)n (−m)n m! , (1.61) Γ (α − n) Γ (α) = (−1)n (1 − α)n , (1.62) Γ (α − n) Γ (α) = (α)−n = (−1)n (1 − α)n , (1.63) n! (α)n+1 − n! (α + 1)n+1 = (n + 1)! (α)n+2 , (1.64) (α)n (β)n − (α)n+1 (β)n+1 = (α)n (β)n+1 (α − β) , (1.65) (α)n−k = (−1)k (1 − α)−n (1 − α − n)k = (−1)k (α)n (1 − α − n)k , (1.66) (1)n−k = (n − k)! = (−1)k (1)n (−n)k = = (−1)k n! (−n)k , (1.67) (α)mn = mmn m j=1 α + j − 1 m n , (1.68) and (−n)k = (−1)kn! (n−k)! , (0 ≤ k ≤ n) 0, (k n) . (1.69) For more details of the results, readers refer to the works [3, 11, 13, 14, 23, 36].
- 34. 12 1 Preliminaries Theorem 1.23 (Stirling [38]) 1 z−α = ∞ k=0 (α)k (z)k+1 = ∞ k=0 Γ (z) Γ (α) Γ (α+k) Γ (z+k+1) = 1 z + α z(z+1) + α(α+1) z(z+1)(z+2) + · · · +, (1.70) where Re (α) 0, Re (z) 0, Re (α − z) 0and α, z ∈ CZ− 0 . The result was discovered by Stirling in 1730 [38] and reviewed by Gronwall in 1916 [11]. Theorem 1.24 Let h(1) (t) 0, h (0) = a, h (1) = b, x ∈ C and Re (x) 0. Then we have Γ (x) = b a e−h(t) (h (t))x−1 h(1) (t) dt . (1.71) The result was discovered by Yang et al. in 2020 when x ∈ N [39]. 1.1.3 The Euler Beta Function In this section, we investigate the concept and theorems of the Euler beta function. Definition 1.5 (Euler [22]) The Euler beta function is defined as B (α, β) = Γ (α) Γ (β) Γ (α + β) = 1 0 tα−1 (1 − t)β−1 dt, (1.72) where Re (α) 0, Re (β) 0, and α, β ∈ CZ− 0 . The formula (called the Euler integral of the second kind) was first discovered by Euler in 1772 [22] and by Legendre in 1811 (see [40], p.211), and the name of the beta function was introduced for the first time by Binet in 1839 [41]. For more details, see the monograph [33]. It is clear that B (α, 1) = 1 α + 1 (1.73)
- 35. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 13 and B (1, β) = 1 1 + β , (1.74) where Re (α) 0, Re (β) 0, and α, β ∈ CZ− 0 . There are some useful formulas as follows: π 2 0 sin t2α−1 cos2β−1 dt = B (α, β) , (1.75) b a (b − t)α−1 (t − a)β−1 dt = B (α, β) (b − a)α+β−1 , (1.76) where α, β ∈ CZ− 0 , a, b ∈ R, Re (α) 0, Re (β) 0 and b a. In the special case of a = 0 and b = x in (1.76), we have x 0 (x − t)α−1 tβ−1 dt = B (α, β) xα+β−1 , (1.77) where α, β ∈ CZ− 0 , x 0 and Re (α) 0, Re (β) 0 x 0. Theorem 1.25 (Whittaker and Watson [20]) Γ (α, n) = lim n→∞ nα B (α, n) , (1.78) where α ∈ CZ− 0 and n ∈ N. The result was reported by Whittaker and Watson (see [20], p.254). Theorem 1.26 (Euler [22]) B (α, β) (α + β, c) = B (β, c) (β + c, α) . (1.79) The result was discovered by Euler in 1772 [22] and further reported by Whittaker and Watson (see [20], p.261). Theorem 1.27 (Whittaker and Watson [20]) B (α, β) = ∞ 0 tα−1 (1 + t)−(α+β) dt, (1.80)
- 36. 14 1 Preliminaries B (α, n + 1) = n! (α)n+1 (n ∈ N0) , (1.81) B (α, β + 1) = β α + β B (α, β) , (1.82) B (α, β) = B (α + 1, β) + B (α, β + 1) , (1.83) B (α, β + 1) = β α B (α + 1, β) , (1.84) and B (α, β) = B (β, α) . (1.85) The results were discovered by Whittaker and Watson (see [20], p.254). Theorem 1.28 (Whittaker and Watson [20]) There is [20] B (nα, nβ) = n−nβ n k=1 B β + n−1 n , α n k=1 B ((k − 1) α, α) , (1.86) where Re (α) 0, Re (β) 0, n ∈ N and α, β ∈ CZ− 0 . Theorem 1.29 (Hankel [42]) 1 Γ (z) = 1 2πi L s−z es ds, s ∈ C, (1.87) where L is the Hankel contour. The result was discovered by Hankel in 1864 [42]. There is an alternative integral representation as follows: Theorem 1.30 (Hankel [42]) 1 Γ (z) = i 2π ℵ (−s)−z e−s ds, s ∈ C, (1.88) where ℵ is the loop contour, which starts at 0i + ∞ encircles the origin and tends to 0i − ∞.
- 37. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 15 The result was discovered by Hankel in 1864 [42]. Definition 1.6 ψ (z) = d log Γ (z) dz = Γ (1) (z) Γ (z) , (1.89) where s ∈ C and log Γ (z) = z − 1 2 log z − s + log √ 2π + Σ (z) (1.90) with the infinite series due to Gudermann [43] Σ (z). The result is the logarithmic derivative of the gamma function [11]. For more information of the Euler gamma function, Pochhammer symbols, and Euler beta function, reader refers to Gronwall [11], Bell [13], Luke [14], Whittaker and Watson [20], Manocha and Srivastava [23], Wang et al. [35], Slater [36], Rainville [37], and Andrews et al. [26]. Theorem 1.31 Let h(1) (t) 0, h (0) = a, h (∞) = b, x, y ∈ C , Re (x) 0 and Re (y) 0. Then we have B (x, y) = b a e−h(t) (h (t))x−1 (1 − h (t))y−1 h(1) (t) dt . (1.91) The result was discovered by Yang et al. in 2020 when x ∈∈ R+ and y ∈ R+ [39]. 1.1.4 The Extended Euler Gamma Functions In this section, we preset the Euler–Chaudhry–Zubair gamma function, which is the extension of the gamma function, and other versions of the gamma-type functions. Definition 1.7 (Chaudhry and Zubair [44]) The Euler–Chaudhry–Zubair gamma function, denoted byI Γp (z), is defined as I Γp (z) = ∞ 0 tz−1 e−t− p t dt, (1.92) where Re (z) 0, Re (p) 0, and z ∈ C. The result was discovered by Chaudhry and Zubair in 1994 [44].
- 38. 16 1 Preliminaries Definition 1.8 The Euler-type gamma function, denoted by II Γp (z), is defined as II Γp (z) = ∞ 0 tz−1 e−t− p t(1−t) dt, (1.93) where Re (z) 0, Re (p) 0, and z ∈ C. In this case, we get II Γp (z) = ∞ 0 tz−1e−t− p t(1−t) dt = ∞ 0 tz−1e−t− p 1−t − p t dt, (1.94) where Re (z) 0, Re (p) 0, and z ∈ C. Definition 1.9 The Euler-type gamma function, denoted by III Γp (z), is defined as III Γp (z) = ∞ 0 tz−1 e−t− p 1−t dt, (1.95) where Re (z) 0, Re (p) 0, and z ∈ C. Definition 1.10 The Euler-type gamma function, denoted by IV Γp,q (z), is defined as IV Γp,q (z) = ∞ 0 tz−1 e−t− p t − q 1−t dt, (1.96) where Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C. Theorem 1.32 If Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C, then we have the following relationships: IV Γp,0 (z) = I Γp (z) , (1.97) IV Γp,p (z) = II Γp (z) , (1.98)
- 39. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 17 IV Γ0,p (z) = III Γp (z) , (1.99) IV Γ0,0 (z) = I Γ0 (z) = II Γ0 (z) = III Γ0 (z) = Γ (z) . (1.100) Definition 1.11 The Euler-type gamma function, denoted by V Γp (z), is defined as V Γp (z) = ∞ 0 tz−1 e−pt dt, (1.101) where Re (z) 0, Re (p) 0 and z ∈ C. Definition 1.12 The Euler-type gamma function, denoted by V I Γp,q,r (z), is defined as V I Γp,q,r (z) = ∞ 0 tz−1 e−rt− p t − q 1−t dt, (1.102) where Re (z) 0, Re (p) 0, Re (q) 0, Re (r) 0, and z ∈ C. Theorem 1.33 If Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C, then we have the following relationships: V I Γ0,0,1 (z) = Γ (z) , (1.103) V I Γp,q,1 (z) = IV Γp,q (z) , (1.104) V I Γ0,0,p (z) = V Γp (z) , (1.105) V I Γ0,p,1 (z) = III Γp (z) , (1.106) V I Γp,p,1 (z) = II Γp (z) , (1.107) and V I Γp,0,1 (z) = I Γp (z) , (1.108) where Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C.
- 40. 18 1 Preliminaries Definition 1.13 The Euler-type gamma function, denoted by V II Γp,r (z), is defined as V II Γp,r (z) = ∞ 0 tz−1 e−rtp dt, (1.109) where Re (z) 0, Re (p) 0, Re (r) 0, and r, z, p ∈ C. Theorem 1.34 If Re (z) 0, Re (p) 0, Re (r) 0, and r, z, p ∈ C, then we have V II Γp,r (z) = r − z q p Γ z p . Corollary 1.1 If Re (z) 0, Re (r) 0, and z, r ∈ C, then we have V II Γ1,r (z) = r−z Γ (z) . (1.110) Corollary 1.2 If Re (z) 0, Re (p) 0, and z, p ∈ C, then we have V II Γp,1 (z) = 1 p Γ z p . (1.111) The results can be obtained from the analytic number theory in Chap. 2. For more details, see [9, 45–47]. 1.1.5 The Extended Euler Beta Functions In this section, we preset the Euler–Chaudhry–Qadir–Rafique–Zubair beta function [48], which is the extension of the beta function, and other related functions. Definition 1.14 The Euler-type beta function, denoted by I Bp (α, β), is defined as I Bp (α, β) = 1 0 tα−1 (1 − t)β−1 e− p t dt, (1.112) where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 .
- 41. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 19 Definition 1.15 (Chaudhry et al. [48]) The Euler–Chaudhry–Qadir–Rafique–Zubair beta function, denoted by II Bp (α, β), is defined as [48] II Bp (α, β) = 1 0 tα−1 (1 − t)β−1 e− p t(1−t) dt, (1.113) where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 . The result was discovered by Chaudhry et al. in 1997 [48]. In this case, we take the form II Bp (α, β) = 1 0 tα−1 (1 − t)β−1 e− p t(1−t) dt = 1 0 tα−1 (1 − t)β−1 e− p 1−t − p t dt, (1.114) where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 . Definition 1.16 The Euler-type beta function, denoted by III Bp (α, β), is defined as III Bp (α, β) = 1 0 tα−1 (1 − t)β−1 e− p 1−t dt, (1.115) where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 . Definition 1.17 The Euler–Choi–Rathie–Parmar beta function, denoted by IV Bp,q (α, β), is defined as [49] IV Bp,q (α, β) = 1 0 tα−1 (1 − t)β−1 e− p t − q 1−t dt, (1.116) where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, and α, β ∈ CZ− 0 . The result was discovered by Choi et al. in 2014 [49].
- 42. 20 1 Preliminaries Definition 1.18 The Euler-type beta function, denoted by V Bp (α, β), is defined as V Bp (α, β) = 1 0 tα−1 (1 − t)β−1 e−pt dt, (1.117) where Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 . Definition 1.19 The Euler-type beta function, denoted by V I Bp,q,r (α, β), is defined as V I Bp,q,r (α, β) = 1 0 tα−1 (1 − t)β−1 e−rt− p t − q 1−t dt, (1.118) where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, Re (r) 0, and α, β ∈ CZ− 0 . Theorem 1.35 Let Re (β) Re (α) 0, Re (p) 0, Re (q) 0, Re (r) 0 and α, β ∈ CZ− 0 . Then we have the following: V I B0,0,p (α, β) = V Bp (α, β) , (1.119) V I Bp,q,0 (α, β) = IV Bp,q (α, β) , (1.120) V I B0,q,0 (α, β) = III Bp (α, β) , (1.121) V I Bp,p,0 (α, β) = II Bp (α, β) , (1.122) V I Bp,0,0 (α, β) = I Bp (α, β) . (1.123) Theorem 1.36 (Chaudhry et al. [50]) Suppose that Re (β) Re (α) 0, Re (γ ) 0, |arg (1 − z)| π, Re (p) 0, and α, β ∈ CZ− 0 , then we have the following integral representation [50] Fp (γ, α, β; z) = 1 B (α, β − α) 1 0 tα−1 (1 − t)β−α−1 (1 − zt)−γ e− p t(1−t) dt. (1.124) The result was discovered by Chaudhry et al. in 2004 [50].
- 43. 1.1 The Euler Gamma Function, Pochhammer Symbols, Euler Beta Function,. . . 21 Theorem 1.37 (Chaudhry–Qadir–Srivastava–Paris Theorem [50]) Suppose that Re (β) Re (α) 0, Re (p) 0, Re (z) 0, and α, β ∈ CZ− 0 , then we have the following integral representation [50] Φp (α, β; z) = 1 B (α, β − α) 1 0 tα−1 (1 − t)β−α−1 e zt− p t(1−t) dt. (1.125) The result was discovered by Chaudhry et al in 2004 [50] As the direct results, we have the following: Corollary 1.3 Suppose that Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 , then we have the following integral representation Φ0 (α, β; z) = 1 B (α, β − α) 1 0 tα−1 (1 − t)β−α−1 ezt dt. (1.126) Proof For the above formula with p = 0, we get the result. Corollary 1.4 Suppose that Re (β) Re (α) 0, Re (p) 0, and α, β ∈ CZ− 0 , then we have Φ0 (α, β; z) = V Bp (α,) B (α, β − α) . (1.127) Proof With the definition, we directly obtain the result. Corollary 1.5 Suppose that Re (β) Re (α) 0, Re (p) 0, Re (z) 0, and α, β ∈ CZ− 0 , then we have the following integral representation Φp (α, β; 0) = II Bp (α, β) B (α, β − α) . (1.128) Proof For z = 0, we have the result. Theorem 1.38 (Choi et al. [49]) Suppose that Re (β) Re (α) 0, p 0, q 0, and α, β ∈ CZ− 0 , then the integral representation 1 0 1 0 IV Bp,q (α, β) pκ−1 qυ−1 dpdq = Γ (κ) Γ (υ) B (α + κ, β + υ) (1.129) holds true. The result was discovered by Choi et al in 2014 [49].
- 44. 22 1 Preliminaries There are the connections among them as follows: Theorem 1.39 If Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C, then we have the following relationships: V I B0,0,1 (z, 0) = B (z, 0) , (1.130) V I Bp,q,1 (z, 0) = IV Bp,q (z, 0) , (1.131) V I B0,0,p (z, 0) = V Bp (z, 0) , (1.132) V I B0,p,1 (z, 0) = III Bp (z, 0) , (1.133) V I Bp,p,1 (z, 0) = II Bp (z, 0) , (1.134) V I Bp,0,1 (z, 0) = I Bp (z, 0) , (1.135) where Re (z) 0, Re (p) 0, Re (q) 0, and z ∈ C. Definition 1.20 The Euler-type beta function, denoted by V II Bp,r (α, β), is defined as V II Bp,r (α, β) = 1 0 tα−1 (1 − t)β−1 e−rtp dt, (1.136) where Re (β) Re (α) 0, Re (p) 0, Re (q) 0, and α, β ∈ CZ− 0 . For the detailed comments of the above formulae, see [45–48]. 1.2 Gauss Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions In this section, we present the definition, theorems, and properties for the Gauss hypergeometric series, and the relationships among the hypergeometric super- trigonometric and superhyperbolic functions via Gauss hypergeometric series. 1.2.1 The Gauss Hypergeometric Series Let us start with the definition of the Gauss hypergeometric series.
- 45. 1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 23 Definition 1.21 (Gauss [27]) The Gauss hypergeometric series of form is defined as [27] 2F1 (a, b; c; z) = 1+ab c z + a(a+1)b(b+1) c(c+1) z2 2 + · · · = ∞ n=0 (a)n(b)n (c)n zn n! , (1.137) where a, b, c, z ∈ C, n ∈ N0, and |z| 1. The formula was discovered by Gauss in 1812 [27]. Theorem 1.40 (Euler [51]) If a, b, c, z ∈ C and |z| 1, then there exists 2F1 (a, b; c; z) = Γ (c) Γ (b) Γ (c − b) 1 0 tb−1 (1 − t)c−b−1 (1 − zt)−a dt, (1.138) where Re (c) Re (b) 0. The result was discovered by Euler in 1679 [51]. Theorem 1.41 (Gauss [27]) If Re (c − a − b) 0, Re (c) Re (a), and Re (c) Re (b), then there exist 2F1 (a, b; c; 1) = ∞ n=0 (a)n (b)n (c)n 1 n! = Γ (c) Γ (c − b − a) Γ (c − a) Γ (c − b) . (1.139) For |z| 1 and a, b, c, z ∈ C the function 2F1 (a, b; c; z) is analytic except for simple poles at c = 0 and c ∈ Z−. The result was discovered by Gauss in 1812 [27]. Theorem 1.42 (Chu-Vandermonde Identity [52, 53]) 2F1 (−n, b; c; 1) = (c − b)n (c)n . (1.140) The result was discovered by Chu in 1303 [52] and by Vandermonde in 1772 [53], respectively. Theorem 1.43 (Bateman [54]) If Re (c) Re (μ) 0 and |z| 1, then there is [54] 2F1 (a, b; c + μ; z) = Γ (c + μ) z1−(c+μ) Γ (c) Γ (μ) z 0 tc−1 (z − t)μ−1 2F1 (a, b; c, t) dt. (1.141) The result was discovered by Bateman in 1909 [54].
- 46. 24 1 Preliminaries Theorem 1.44 (Bateman [54]) If Re (c) Re (μ) 0 and |z| 1, then there is [54] 2F1 (a, b; c + μ; z) = Γ (c + μ) Γ (c) Γ (μ) 1 0 tc−1 (1 − t)μ−1 2F1 (a, b; c, zt) dt. (1.142) This is the special case of Bateman [54]. Theorem 1.45 (Koshliakov [55]) If Re (c) Re (μ) 0 and |z| 1, then there is 1F1 (a, b; c + μ; z) = Γ (c + μ) Γ (c) Γ (μ) 1 0 tc−1 (1 − t)μ−1 1F1 (a, b; c, zt) dt. (1.143) This result is known as the Koshliakov’s formula [55]. Property 1.1 The hypergeometric series is a solution of the Gauss differential equation of form z (1 − z) d2ϕ dz2 + (c − (a + b + 1) z) dϕ dz − abϕ=0, (1.144) where a, b, z ∈ C , c ∈ CZ− 0 , and |z| 1. The result was reported by Whittaker and Watson in 1927 [20], and by Bailey in 1935 [56]. Theorem 1.46 (Barnes [57]) Suppose that Re (c) Re (μ) 0, |z| 1, and |arg (−z)| π, then we have [57] 2F1 (a, b; c; z) = Γ (c) Γ (a) Γ (b) 1 2πi i∞ −i∞ Γ (a + s) Γ (b + s) Γ (−s) Γ (c + s) (−z)s ds. (1.145) The result was reported by Barnes in 1908 [57]. Some applications of the ODEs in mechanics of shells were presented by Sun in [58–61].
- 47. 1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 25 1.2.2 The Hypergeometric Supertrigonometric Functions via Gauss Superhyperbolic Series We now consider the hypergeometric supertrigonometric functions via Gauss superhyperbolic series [39]. Definition 1.22 The hypergeometric supersine via Gauss superhyperbolic series is defined as [39] 2Supersin1 (a, b; c; z) = ∞ n=0 (a)2n+1 (b)2n+1 (c)2n+1 (−1)n z2n+1 (2n + 1)! , (1.146) where a, b, c, z ∈ C. Definition 1.23 The hypergeometric supercosine via Gauss superhyperbolic series is defined as [39] 2Supercos1 (a, b; c; z) = ∞ n=0 (a)2n (b)2n (c)2n (−1)n z2n (2n)! , (1.147) where a, b, c, z ∈ C. Definition 1.24 The hypergeometricsupertangent via Gauss superhyperbolicseries is defined as [39] 2Supertan1 (a, b; c; z) = 2Supersin1 (a, b; c; z) 2Supercos1 (a, b; c; z) , (1.148) where a, b, c, z ∈ C. Definition 1.25 The hypergeometric supercotangent via Gauss superhyperbolic series is defined as [39] 2Supercot1 (a, b; c; z) = 2Supercos1 (a, b; c; z) 2Supersin1 (a, b; c; z) , (1.149) where a, b, c, z ∈ C. Definition 1.26 The hypergeometric supersecant via Gauss superhyperbolic series is defined as [39] 2Supersec1 (a, b; c; z) = 1 2Supercos1 (a, b; c; z) , (1.150) where a, b, c, z ∈ C.
- 48. 26 1 Preliminaries Definition 1.27 The hypergeometric supercosecant via Gauss superhyperbolic series is defined as [39] 2Supercosec1 (a, b; c; z) = 1 2Supersin1 (a, b; c; z) , (1.151) where a, b, c, z ∈ C. The formulae for the hypergeometric supertrigonometric functions via Gauss superhyperbolic series were discovered by Yang in 2020 [39]. In this case, we present [39] 2Supersin1 (a, b; c; λz) = 1 2i (2F1 (a, b; c; iλz) −2 F1 (a, b; c; −iλz)) = ∞ n=0 (a)2n+1(b)2n+1 (c)2n+1 (−1)n(λz)2n+1 (2n+1)! , (1.152) 2Supercos1 (a, b; c; λz) = 1 2 (2F1 (a, b; c; iλz) +2 F1 (a, b; c; −iλz)) = ∞ n=0 (a)2n(b)2n (c)2n (−1)n(λz)2n (2n)! , (1.153) 2Supertan1 (a, b; c; λz) = 2Supersin1 (a, b; c; λz) 2Supercos1 (a, b; c; λz) , (1.154) 2Supercot1 (a, b; c; λz) = 2Supercos1 (a, b; c; λz) 2Supersin1 (a, b; c; λz) , (1.155) 2Supersec1 (a, b; c; λz) = 1 2Supercos1 (a, b; c; λz) , (1.156) and 2Supercosec1 (a, b; c; λz) = 1 2Supersin1 (a, b; c; λz) , (1.157) where a, b, c, λ, z ∈ C. Theorem 1.47 Suppose that a, b, c, z ∈ C, and i = √ −1, then we have 2F1 (a, b; c; iz) = 2Supercos1 (a, b; c; z) + i2Supersin1 (a, b; c; z) . (1.158) The result was discovered by Yang in 2020 [39]. More generally, we have the following formula: Suppose that a, b, c, λ, z ∈ C, and i = √ −1, then we have [39] 2F1 (a, b; c; iλz) = 2Supercos1 (a, b; c; λz) + i2Supersin1 (a, b; c; λz) . (1.159)
- 49. 1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 27 1.2.3 The Hypergeometric Superhyperbolic Functions via Gauss Hypergeometric Series We now investigate the hypergeometric superhyperbolic functions via Gauss hyper- geometric series. Definition 1.28 The hypergeometric superhyperbolic sine via Gauss hypergeomet- ric series is defined as [39] 2Supersinh1 (a, b; c; z) = ∞ n=0 (a)2n+1 (b)2n+1 (c)2n+1 z2n+1 (2n + 1)! , (1.160) where a, b, c, z ∈ C. Definition 1.29 The hypergeometric superhyperbolic cosine via Gauss hypergeo- metric series is defined as [39] 2Supercosh1 (a, b; c; z) = ∞ n=0 (a)2n (b)2n (c)2n z2n (2n)! , (1.161) where a, b, c, z ∈ C. Definition 1.30 The hypergeometric superhyperbolic tangent via Gauss hypergeo- metric series is defined as [39] 2Supertanh1 (a, b; c; z) = 2Supersinh1 (a, b; c; z) 2Supercosh1 (a, b; c; z) , (1.162) where a, b, c, z ∈ C. Definition 1.31 The hypergeometric superhyperbolic cotangent via Gauss hyper- geometric series is defined as [39] 2Supercoth1 (a, b; c; z) = 2Supercosh1 (a, b; c; z) 2Supersinh1 (a, b; c; z) , (1.163) where a, b, c, z ∈ C. Definition 1.32 The hypergeometric superhyperbolic secant via Gauss hypergeo- metric series is defined as [39] 2Supersech1 (a, b; c; z) = 1 2Supercosh1 (a, b; c; z) , (1.164) where a, b, c, z ∈ C.
- 50. 28 1 Preliminaries Definition 1.33 The hypergeometric superhyperbolic cosecant via Gauss hyperge- ometric series is defined as [39] 2Supercosech1 (a, b; c; z) = 1 2Supersinh1 (a, b; c; z) , (1.165) where a, b, c, z ∈ C. The results were proposed by Yang in 2020 [39]. So, there exist [39] 2Supersinh1 (a, b; c; λz) = 1 2 (2F1 (a, b; c; λz) −2 F1 (a, b; c; −λz)) = ∞ n=0 (a)2n+1(b)2n+1 (c)2n+1 (λz)2n+1 (2n+1)! , (1.166) 2Supercosh1 (a, b; c; λz) = 1 2 (2F1 (a, b; c; λz) +2 F1 (a, b; c; −λz)) = ∞ n=0 (a)2n(b)2n (c)2n (λz)2n (2n)! , (1.167) 2Supertanh1 (a, b; c; λz) = 2Supersinh1 (a, b; c; λz) 2Supercosh1 (a, b; c; λz) , (1.168) 2Supercoth1 (a, b; c; λz) = 2Supercosh1 (a, b; c; λz) 2Supersinh1 (a, b; c; λz) , (1.169) 2Supersech1 (a, b; c; λz) = 1 2Supercosh1 (a, b; c; λz) , (1.170) 2Supercosech1 (a, b; c; λz) = 1 2Supersinh1 (a, b; c; λz) , (1.171) where a, b, c, λ, z ∈ C. Theorem 1.48 Suppose that a, b, c, z ∈ C, then we have [39] 2F1 (a, b; c; z) = 2Supercosh1 (a, b; c; z)+2Supersinh1 (a, b; c; z) . (1.172) So, we have [39] 2F1 (a, b; c; λz) = 2Supercosh1 (a, b; c; λz) + 2Supersinh1 (a, b; c; λz) , (1.173) where a, b, c, λ, z ∈ C.
- 51. 1.2 Gauss Hypergeometric Series and Supertrigonometric and. . . 29 Theorem 1.49 Let a, b, c ∈ C. Then we have [39] 2Supersinh1 (a, b; c; 0) = 0. (1.174) Theorem 1.50 Let a, b, c ∈ C. Then we have [39] 2Supercosh1 (a, b; c; 0) = 1. (1.175) Theorem 1.51 Let a, b, c ∈ C. Then we have [39] 2Supertanh1 (a, b; c; 0) = 0. (1.176) Theorem 1.52 Let a, b, c ∈ C. Then we have [39] 2Supersec1 (a, b; c; 0) = 1. (1.177) The results were discovered by Yang in 2020 [39]. Theorem 1.53 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0, Re (c1) 0, s ∈ N, and |z| 1, then we have [39] z 0 tα−1 (z − t)β−1 2F1 a1, a2; c1; λ (z − t)s dt = B (α, β) zα+β−1 ×2+s F1+s a1, a2, β s , β+1 s , · · · , β+s−1 s ; c1, α+β s , α+β+1 s , · · · , α+β+s−1 s ; λzs , (1.178) where λ is a constant. Theorem 1.54 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (c1) 0,Re (c2) 0, s ∈ N, and |z| 1, then we have [39] z 0 tα−1 (z − t)β−1 1F2 a1; c1, c2; λ (z − t)s dt = B (α, β) zα+β−1 1+sF2+s a1, β s , β+1 s , · · · , β+s−1 s ; c1, c2, α+β s , α+β+1 s , · · · , α+β+s−1 s ; λzs , (1.179) where λ is a constant.
- 52. 30 1 Preliminaries Theorem 1.55 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0, Re (c1) 0, and |z| 1, then we have [39] z 0 tα−1 (z − t)β−1 2F1 (a1, a2; c1; λ (z − t)) dt = B (α, β) zα+β−1 3F2 a1, a2, β; c1, α + β; λz , (1.180) where λ is a constant and α, β, a1, a2, c1 ∈ C. Theorem 1.56 Suppose that Re (α) 0, Re (β) 0, Re (a1) 0, Re (a2) 0, Re (c1) 0, and s ∈ N, then we have [39] 1 −1 (t + 1)α−1 (1 − t)β−1 2F1 a1, a2; c1; λ (1 − t)s dt = B (α, β) 2α+β−1 2+sF1+s a1, a2, β s , β+1 s , · · · , β+s−1 s ; c1, α+β s , α+β+1 s , · · · , α+β+s−1 s ; λ2s , (1.181) where λ is a constant and α, β, a1, a2, c1 ∈ C. For more details of the applications of the above results, see [39, 62–67]. 1.3 Clausen Hypergeometric Series and Supertrigonometric and Superhyperbolic Functions In this section, we introduce the definition, theorems, and properties for the Clausen hypergeometric series. 1.3.1 The Clausen Hypergeometric Series We now consider the definition and theorems for the Clausen hypergeometric series. Definition 1.34 (Clausen [68]) The Clausen hypergeometric series of form is defined as [68] pFq ((a) , (c) ; z) =p Fq a1, · · · , ap; c1, · · · , cq; z =p Fq a1, · · · , ap c1, · · · , cq ; z = ∞ n=0 (a1)n···(ap)n (c1)n···(cq)n zn n! , (1.182) where an, cn, z ∈ C and n, p, q ∈ N0.
- 53. 1.3 Clausen Hypergeometric Series and Supertrigonometric and. . . 31 The result, as an extended version of the Gauss hypergeometric series, was discovered by Clausen in 1828 [68]. Theorem 1.57 (Convergences for the Clausen Hypergeometric Series) The cases of the convergences of the Clausen hypergeometric series (1.182) hold for an ∈ CZ− 0 : (1) if p q, then the series converges absolutely for z ∈ C; (2) if p = q + 1, then the series converges absolutely for |z| 1 and diverges for |z| 1, and for |z| = 1 it converges absolutely for Re q k=1 ck − p k=1 ak 0; (3) if p q + 1, then the series converges only for z = 0. The result was reported by Bailey [56], Srivastava and Kashyap [69], Slater [36], Andrews et al. [26], and Rainville [37]. Theorem 1.58 (The Differential Equation for the Clausen Hypergeometric Series) If an, cn, z ∈ C, n, p, q ∈ N0, and |z| 1, the Clausen hypergeometric series is a solution of the differential equation [69] (Q (q, μ, cn) ϕ) (z) − (P (p, μ, an) ϕ) (z) = 0, (1.183) where (Q (q, μ, cn) ϕ) (z) = z d dz q n=1 z d dz ϕ (z) + (cn − 1) ϕ (z) = z d dz q n=1 z d dz + (cn − 1) ϕ (z) (1.184) and (P (p, μ, an) ϕ) (z) = z p n=1 zdϕ(z) dz + anϕ (z) = z p n=1 z d dz + an ϕ (z). (1.185) The result was reported by Srivastava and Kashyap [69], Rainville [37], Andrews et al. [26], and Luke [14].
- 54. 32 1 Preliminaries Theorem 1.59 (The Derivative of the Clausen Hypergeometric Series) If where an, cn, λ, z ∈ C, n, p, q ∈ N0, Re q k=1 ck − p k=1 ak 0, and |z| 1, then there is the derivative of the Clausen hypergeometric series as follows [69]: d dz pFq a1, · · · , ap; c1, · · · , cq; z = p n=1 an q n=1 cn pFq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; z (1.186) and 1 λ d dz pFq a1, · · · , ap; c1, · · · , cq; λz = p n=1 an q n=1 cn pFq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; λz . (1.187) The result was discussed by Srivastava and Kashyap [69], Rainville [37], Andrews et al. [26], and Luke [14]. Theorem 1.60 (Rainville [37]) If p ≤ q + 1, Re (a1) 0,· · · , Re ap 0, Re (c1) 0, · · · , Re cq 0 , Re q k=1 ck − p k=1 ak 0and |z| 1, then there is [37] pFq a1, · · · , ap; c1, · · · , cq; λz = Γ (c1) Γ (a1)Γ (c1−a1) 1 0 ta1−1 (1 − t)c1−a1−1 p−1Fq−1 a2, · · · , ap; c2, · · · , cq; λzt dt, (1.188) where λ is a constant. The result was discovered by Rainville in 1960 (see [37], p.85). Theorem 1.61 (Askey [70], p.19) If p ≤ q + 1, Re (a1) 0,· · · , Re ap 0, Re (c1) 0, · · · , Re cq 0 , Re q k=1 ck − p k=1 ak 0and |z| 1, then there is [70] p+1Fq+1 a1, · · · , ap, a; c1, · · · , cq, c; λz = Γ (c) Γ (a)Γ (c−a) 1 0 ta−1 (1 − t)c−a−1 pFq a1, a2, · · · , ap; c1, c2, · · · , cq; λzt dt, (1.189) where λ is a constant.
- 55. Another Random Scribd Document with Unrelated Content
- 56. been discerned by this sulphurous light; but my whole attention was absorbed by the river, which seemed to come out of the darkness like an apparition at the summons of my impatient will. It could be borne only for a short time; this dazzling, bewildering alternation of glare and blackness, of vast reality and nothingness. I was soon glad to draw back from the precipice and seek the candlelight within. The next day was Sunday. I shall never forget, if I live to a hundred, how the world lay at my feet one Sunday morning. I rose very early, and looked abroad from my window, two stories above the platform. A dense fog, exactly level with my eyes, as it appeared, roofed in the whole plain of the earth; a dusky firmament in which the stars had hidden themselves for the day. Such is the account which an antediluvian spectator would probably have given of it. This solid firmament had spaces in it, however, through which gushes of sunlight were poured, lighting up the spires of white churches, and clusters of farm buildings too small to be otherwise distinguished; and especially the river, with its sloops floating like motes in the sunbeam. The firmament rose and melted, or parted off into the likeness of snowy sky-mountains, and left the cool Sabbath to brood brightly over the land. What human interest sanctifies a bird's-eye view! I suppose this is its peculiar charm, for its charm is found to deepen in proportion to the growth of mind. To an infant, a champaign of a hundred miles is not so much as a yard square of gay carpet. To the rustic it is less bewitching than a paddock with two cows. To the philosopher, what is it not? As he casts his eye over its glittering towns, its scattered hamlets, its secluded homes, its mountain ranges, church spires, and untrodden forests, it is a picture of life; an epitome of the human universe; the complete volume of moral philosophy, for which he has sought in vain in all libraries. On the left horizon are the Green Mountains of Vermont, and at the right extremity sparkles the Atlantic. Beneath lies the forest where the deer are hiding and the birds rejoicing in song. Beyond the river he sees spread the rich plains of Connecticut; there, where a blue expanse lies beyond the triple range of hills, are the churches of religious Massachusetts sending up their Sabbath psalms; praise which he is too high to hear, while God is not. The fields and waters seem to him to-day no more truly property than the skies which shine down upon them; and to think how some below are busying their thoughts this Sabbath-day about how they shall hedge in another field, or multiply their flocks on yonder meadows, gives
- 57. him a taste of the same pity which Jesus felt in his solitude when his followers were contending about which should be greatest. It seems strange to him now that man should call anything his but the power which is in him, and which can create somewhat more vast and beautiful than all that this horizon encloses. Here he gains the conviction, to be never again shaken, that all that is real is ideal; that the joys and sorrows of men do not spring up out of the ground, or fly abroad on the wings of the wind, or come showered down from the sky; that good cannot be hedged in, nor evil barred out; even that light does not reach the spirit through the eye alone, nor wisdom through the medium of sound or silence only. He becomes of one mind with the spiritual Berkeley, that the face of nature itself, the very picture of woods, and streams, and meadows, is a hieroglyphic writing in the spirit itself, of which the retina is no interpreter. The proof is just below him (at least it came under my eye), in the lady (not American) who, after glancing over the landscape, brings her chair into the piazza, and, turning her back to the champaign, and her face to the wooden walls of the hotel, begins the study, this Sunday morning, of her lapful of newspapers. What a sermon is thus preached to him at this moment from a very hackneyed text! To him that hath much; that hath the eye, and ear, and wealth of the spirit, shall more be given; even a replenishing of this spiritual life from that which to others is formless and dumb; while from him that hath little, who trusts in that which lies about him rather than in that which lives within him, shall be taken away, by natural decline, the power of perceiving and enjoying what is within his own domain. To him who is already enriched with large divine and human revelations this scene is, for all its stillness, musical with divine and human speech; while one who has been deafened by the din of worldly affairs can hear nothing in this mountain solitude. The march of the day over the valley was glorious, and I was grieved to have to leave my window for an expedition to the Falls a few miles off. The Falls are really very fine, or, rather, their environment; but I could see plenty of waterfalls elsewhere, but nowhere else such a mountain platform. However, the expedition was a good preparation for the return to my window. The little nooks of the road, crowded with bilberries, cherries, and alpine plants, and the quiet tarn, studded with golden water-lilies, were a wholesome contrast to the grandeur of what we had left behind us.
- 58. On returning, we found dinner awaiting us, and also a party of friends out of Massachusetts, with whom we passed the afternoon, climbing higher and higher among the pines, ferns, and blue-berries of the mountain, to get wider and wider views. They told me that I saw Albany, but I was by no means sure of it. This large city lay in the landscape like an anthill in a meadow. Long before sunset I was at my window again, watching the gradual lengthening of the shadows and purpling of the landscape. It was more beautiful than the sunrise of this morning, and less so than that of the morrow. Of this last I shall give no description, for I would not weary others with what is most sacred to me. Suffice it that it gave me a vivid idea of the process of creation, from the moment when all was without form and void, to that when light was commanded, and there was light. Here, again, I was humbled by seeing what such things are to some who watch in vain for what they are not made to see. A gentleman and lady in the hotel intended to have left the place on Sunday. Having overslept that morning's sunrise, and arrived too late for that on Saturday, they were persuaded to stay till Monday noon; and I was pleased, on rising at four on Monday morning, to see that they were in the piazza below, with a telescope. We met at breakfast, all faint with hunger, of course. Well, Miss M., said the gentleman, discontentedly, I suppose you were disappointed in the sunrise. No, I was not. Why, do you think the sun was any handsomer here than at New-York? I made no answer; for what could one say? But he drove me by questions to tell what I expected to see in the sun. I did not expect to see the sun green or blue. What did you expect, then? I was obliged to explain that it was the effect of the sun on the landscape that I had been looking for. Upon the landscape! Oh! but we saw that yesterday.
- 59. The gentleman was perfectly serious; quite earnest in all this. When we were departing, a foreign tourist was heard to complain of the high charges! High charges! As if we were to be supplied for nothing on a perch where the wonder is if any but the young ravens get fed! When I considered what a drawback it is in visiting mountain-tops that one is driven down again almost immediately by one's bodily wants, I was ready to thank the people devoutly for harbouring us on any terms, so that we might think out our thoughts, and compose our emotions, and take our fill of that portion of our universal and eternal inheritance.
- 60. WEDDINGS. God, the best maker of all marriages, Combine your hearts in one! Henry V. I was present at four weddings in the United States, and at an offer of marriage. The offer of marriage ought hardly to be so called, however. It was a petition from a slave to be allowed to wed (as slaves wed) the nursemaid of a lady in whose house I was staying. The young man could either write a little, or had employed some one who could to prepare his epistle for him. It ran from corner to corner of the paper, which was daubed with diluted wafer, like certain love-letters nearer home than Georgia. Here are the contents: Miss Cunningham it is My wishes to companion in your Present and I hope you will Be peeze at it and I hope that you will not think Hard of Me I have Ben to the Doctor and he was very well satafide with Me and I hope you is and Miss Mahuw all so thats all I has to say now wiheshen you will grant Me that honour I will Be very glad. S.B. Smith.
- 61. The nursemaid was granted; and as it was a love-match, and as the girl's mistress is one of the tender, the sore-hearted about having slaves, I hope the poor creatures are as happy as love in debasement can make them. The first wedding I saw in Boston was very like the common run of weddings in England. It happened to be convenient that the parties should be married in church; and in the Unitarian church in which they usually worshipped we accordingly awaited them. I had no acquaintance with the family, but went on the invitation of the pastor who married them. The family connexion was large, and the church, therefore, about half full. The form of celebration is at the pleasure of the pastor; but, by consent, the administration by pastors of the same sect is very nearly alike. The promises of the married parties are made reciprocal, I observed. The service in this instance struck me as being very beautiful from its simplicity, tenderness, and brevity. There was one variation from the usual method, in the offering of one of the prayers by a second pastor, who, being the uncle of the bridegroom, was invited to take a share in the service. The young people were to set out for Europe in the afternoon, the bride being out of health, the dreary drawback upon almost every extensive plan of action and fair promise of happiness in America. The lady has, I rejoice to hear, been quite restored by travel; but her sickness threw a gloom over the celebration, even in the minds of strangers. She and her husband walked up the middle aisle to the desk where the pastors sat. They were attended by only one bridesmaid and one groomsman, and were all in plain travelling dresses. They said steadily and quietly what they had to say, and walked down the aisle again as they came. Nothing could be simpler and better, for this was not a marriage where festivity could have place. If there is any natural scope for joy, let weddings by all means be joyous; but here there was sickness, with the
- 62. prospect of a long family separation, and there was most truth in quietness. The other wedding I saw in Boston was as gay a one as is often seen. The parties were opulent, and in the first rank in society. They were married in the drawing-room of the bride's house, at half past eight in the evening, by Dr. Channing. The moment the ceremony was over, crowds of company began to arrive; and the bride, young and delicate, and her maidens, were niched in a corner of one of the drawing-rooms to courtesy to all comers. They were so formally placed, so richly and (as it then seemed) formally dressed, for the present revived antique style of dress was then quite new, that, in the interval of their courtesies, they looked like an old picture brought from Windsor Castle. The bride's mother presided in the other drawing-room, and the bridegroom flitted about, universally attentive, and on the watch to introduce all visiters to his lady. The transition from the solemnity of Dr. Channing's service to the noisy gayeties of a rout was not at all to my taste. I imagined that it was not to Dr. Channing's either, for his talk with me was on matters very little resembling anything that we had before our eyes; and he soon went away. The noise became such as to silence all who were not inured to the gabble of an American party, the noisiest kind of assemblage, I imagine (not excepting a Jew's synagogue), on the face of the globe. I doubt whether any pagans in their worship can raise any hubbub to equal it. I constantly found in a large party, after trying in vain every kind of scream that I was capable of, that I must give up, and satisfy myself with nodding and shaking my head. If I was rightly understood, well and good; if not, I must let it pass. As the noise thickened and the heat grew more oppressive, I glanced towards the poor bride in her corner, still standing, still courtesying; her pale face growing paler; her nonchalant manner (perhaps the best she could assume) more indifferent. I was afraid that if all this went on much longer, she would faint or die upon the spot. It did not last much longer. By eleven some of the company began to go away, and by a quarter before twelve all
- 63. were gone but the comparatively small party (including ourselves) who were invited to stay to supper. The chandelier and mantelpieces, I then saw, were dressed with flowers. There was a splendid supper; and, before we departed, we were carried up to a well-lighted apartment, where bride cake and the wedding presents were set out in bright array. Five days afterward we went, in common with all her acquaintance, to pay our respects to the bride. The courtyard of her mother's house was thronged with carriages, though no one seemed to stay five minutes. The bridegroom received us at the head of the stairs, and led us to his lady, who courtesied as before. Cake, wine, and liqueurs were handed round, the visiters all standing. A few words on common subjects were exchanged, and we were gone to make way for others. A Quaker marriage which I saw at Philadelphia was scarcely less showy in its way. It took place at the Cherrystreet church, belonging to the Hicksites. The reformed Quaker Church, consisting of the followers of Elias Hicks, bears about the same relation to the old Quakerism as the Church of England to that of Rome; and, it seems to me, the mutual dislike is as intense. I question whether religious enmity ever attained a greater extreme than among the orthodox Friends of Philadelphia. The Hicksites are more moderate, but are sometimes naturally worried out of their patience by the meddling, the denunciations, and the calumnies of the old Quaker societies. The new church is thinking of reforming and relaxing a good deal farther, and in the celebration of marriage among other things. It is under consideration (or was when I was there) whether the process of betrothment should not be simplified, and marriage in the father's house permitted to such as prefer it to the church. The wedding at which I was present was, however, performed with all the formalities.
- 64. A Quaker friend of mine, a frequent preacher, suggested, a few days previously, that a seat had better be reserved for me near the speakers, that I might have a chance of hearing in case there should be communications. I had hopes from this that my friend would speak, and my wishes were not disappointed. The spacious church was crowded; and for three or four hours the poor bride had to sit facing the assemblage, aware, doubtless, that during the time of silence the occupation of the strangers present, if not of the friends themselves, would be watching her and her party. She was pretty, and most beautifully dressed. I have seldom pitied anybody more than I did her, while she sat palpitating for three hours under the gaze of some hundreds of people; but, towards the end of the time of silence, my compassion was transferred to the bridegroom. For want of something to do, after suppressing many yawns, he looked up to the ceiling; and in the midst of an empty stare, I imagine he caught the eye of an acquaintance in the back seats; for he was instantly troubled with a most irrepressible and unseasonable inclination to laugh. He struggled manfully with his difficulty; but the smiles would come, broader and broader. If, by dint of looking steadfastly into his hat for a few minutes, he attained a becoming gravity, it was gone the moment he raised his head. I was in a panic lest we should have a scandalous peal of merriment if something was not given him to do or listen to. Happily there were communications, and the course of his ideas was changed. Of the five speakers, one was an old gentleman whose discourse was an entire perplexity to me. For nearly an hour he discoursed on Jacob's ladder; but in a style so rambling, and in a chant so singularly unmusical as to set attention and remembrace at defiance. Some parenthetical observations alone stood a chance of being retained, from their singularity; one, for instance, which he introduced in the course of his narrative about Jacob setting a stone for a pillow; a very different, cried the preacher, raising his chant to the highest pitch, a very different pillow, by-the-way,
- 65. from any that we—are—accommodated—with. What a contrast was the brief discourse of my Quaker friend which followed! Her noble countenance was radiant as the morning; her soft voice, though low, so firm that she was heard to the farthest corner, and her little sermon as philosophical as it was devout. Send forth thy light and thy truth, was her text. She spoke gratefully of intellectual light as a guide to spiritual truth, and anticipated and prayed for an ultimate universal diffusion of both. The certificate of the marriage was read by Dr. Parrish, an elderly physician of Philadelphia, the very realization of all my imaginings of the personal appearance of William Penn; with all the dignity and bonhommie that one fancies Penn invested with in his dealings with the Indians. Dr. Parrish speaks with affection of the Indians, from the experience some ancestors of his had of the hospitality of these poor people when they were in a condition to show hospitality. His grandfather's family were shipwrecked, and the Indians took the poor lady and her children home to an inhabited cave, and fed them for many weeks or months. The tree stump round which they used to sit at meals is still standing; and Dr. Parrish says that, let it stand as long as it will, the love of his family to the Indians shall outlast it. The matrimonial promise was distinctly and well spoken by both the parties. At the request of the bride and bridegroom, Dr. Parrish asked me to put the first signature, after their own, to the certificate of the marriage; and we adjourned for the purpose to an apartment connected with the church. Most ample sheets of parchment were provided for the signatures; and there was a prodigious array of names before we left, when a crowd was still waiting to testify. This multitudinous witnessing is the pleasantest part of being married by acclamation. If weddings are not to be private, there seems no question of the superiority of this Quaker method to that of the Boston marriage I beheld, where there was all the publicity, without the co-operation and sanction.
- 66. The last wedding which I have to give an account of is full of a melancholy interest to me now. All was so joyous, so simple, so right, that there seemed no suggestion to evil-boding, no excuse for anticipating such wo as has followed. On one of the latter days of July, 1835, I reached the village of Stockbridge; the Sedgwicks' village, for the second time, intending to stay four or five days with my friends there. I had heard of an approaching wedding in the family connexion, and was glad that I had planned to leave, so as to be out of the way at a time when I supposed the presence of foreigners, though friends, might be easily dispensed with. But when Miss Sedgwick and I were sitting in her room one bright morning, there was a tap at the door. It was the pretty black-eyed girl who was to be married the next week. She stood only a minute on the threshold to say, with grave simplicity, I am come to ask you to join our friends at my father's house next Tuesday evening. Being thus invited, I joyfully assented, and put off my journey. The numerous children of the family connexion were in wild spirits all that Tuesday. In the morning we went a strong party to the Ice Hole; a defile between two hills, so perplexed and encumbered with rocks that none but practised climbers need attempt the passage. It was a good way for the young people to work off their exuberant spirits. Their laughter was heard from amid the nooks and hiding-places of the labyrinth, and smiling faces might be seen behind every shrubby screen which sprang up from the crevices. How we tried to surpass each other in the ferns and mosses we gathered, rich in size and variety! What skipping and scrambling there was; what trunk bridges and ladders of roots! How valiant the ladies looked with their stout sticks! How glad every one was to feast upon the wild raspberries when we struggled through the close defile into the cool, green, breezy meadow on the banks of the Housatonic! During the afternoon we were very quiet, reading one of Carlyle's reviews aloud (for the twentieth time, I believe, to some of the party), and discussing it and other things. By eight o'clock we were all dressed for the wedding; and some of the
- 67. children ran over the green before us, but came back, saying that all was not quite ready; so we got one of the girls to sing to us for another half hour. The house of the bride's father was well lighted, and dressed with flowers. She had no mother; but her elder sisters aided her father in bidding us welcome. The drawing-room was quite full; and while the grown-up friends found it difficult to talk, and to repress the indefinable anxiety and agitation which always attend a wedding, the younger members of the party were amusing themselves with whispered mirth. The domestics looked as if the most joyous event of their lives were taking place, and the old father seemed placid and satisfied. In a few minutes we were summoned to another room, at the top of which stood the tall bridegroom, with his pretty little lady on his arm; on either side, the three gentlemen and three ladies who attended them; and in front, the Episcopalian minister who was to marry them, and who has since been united to one of the sisters. It was the first time of his performing the ceremony, and his manner was solemn and somewhat anxious, as might be expected. The bridegroom was a professor in a college in the neighbouring State of New-York; a young man of high acquirements and character, to whom the old father might well be proud to give his daughter. His manners were remarkably pleasing; and there was a joyous, dignified serenity visible in them this evening, which at once favourably prepossessed us who did not previously know him. He was attended by a brother professor from the same college. When the service was over, we all kissed the grave and quiet bride. I trust that no bodings of the woes which awaited her cast a shadow over her spirits then. I think, though grave, she was not sad. She spoke with all her father's guests in the course of the evening, as did her husband. How often have I of late tried to
- 68. recall precisely what they said to me, and every look with which they said it! We went back to the drawing-room for cake and wine; and then ensued the search for the ring in the great wedding cake, with much merriment among those who were alive to all the fun of a festivity like this, and to none of the care. There was much moving about between the rooms, and dressing with flowers in the hall; and lively conversation, as it must needs be where there are Sedgwicks. Then Champagne and drinking of healths went round, the guests poured out upon the green, all the ladies with handkerchiefs tied over their heads. There we bade good-night, and parted off to our several homes. When I left the village the next morning two or three carriages full of young people were setting off, as attendants upon the bride and bridegroom, to Lebanon. After a few such short excursions in the neighbourhood, the young couple went home to begin their quiet college and domestic life. Before a year had elapsed, a year which to me seemed gone like a month, I was at Stockbridge again and found the young wife's family in great trouble. She was in a raging fever, consequent on her confinement, and great fears were entertained for her life. Her infant seemed to have but a small chance under the circumstances, and there was a passing mention of her husband being ill. Every one spoke of him with a respect and affection which showed how worthy he was of this young creature's love; and it was our feeling for him which made our prayers for her restoration so earnest as they were. The last I heard of her before I left the country was that she was slowly and doubtfully recovering, but had not yet been removed from her father's house. The next intelligence that I received after my return to England was of her husband's death; that he had died in a calm and satisfied state of mind; satisfied that if their reasonable hopes of domestic joy and usefulness had not been fulfilled, it was for wise
- 69. and kind reasons; and that the strong hand which thus early divided them would uphold the gentle surviver. No one who beheld and blessed their union can help beseeching and trusting, since all other hope is over, that it may be even thus.
- 70. HIGH ROAD TRAVELLING. How far my pen has been fatigued like those of other travellers in this journey of it, the world must judge; but the traces of it, which are now all set o'vibrating together this moment, tell me it is the most fruitful and busy period of my life; for, as I had made no convention with my man with the gun as to time—by seizing every handle, of what size or shape soever which chance held out to me in this journey—I was always in company, and with great variety too.—Sterne. Our first land travelling, in which we had to take our chance with the world in general, was across the State of New-York. My account of what we saw may seem excessively minute in some of its details; but this style of particularity is not adopted without reasons. While writing my journal, I always endeavoured to bear in mind the rapidity with which civilization advances in America, and the desirablness of recording things precisely in their present state, in order to have materials for comparison some few years hence, when travelling may probably be as unlike what it is now, as a journey from London to Liverpool by the new railroad differs from the same enterprise as undertaken a century and a half ago. To avoid some of the fatigues and liabilities of common travelling, certain of our shipmates and their friends and ourselves had made up a party to traverse the State of New-York in an exclusive
- 71. extra; a stage hired, with the driver, for our own use, to proceed at our own time. Our fellow-travellers were a German and a Dutch gentleman, and the Prussian physician and young South Carolinian whom I have mentioned in the list of our shipmates. We were to meet at the Congress Hall hotel in Albany on the 6th of October. On our way from Stockbridge to Albany we saw a few objects characteristic of the country. While the horses were baiting we wandered into a graveyard, where the names on the tombstones were enough to inform any observer what country of the world he was in. One inscription was laudatory of Nelson and Nabby Bullis; another of Amasa and Polly Fielding. Hiram and Keziah were there too. The signs in the American streets are as ludicrous for their confusion of Greek, Roman, and Hebrew names, as those of Irish towns are for the arbitrary divisions of words. One sees Rudolphus figuring beside Eliakim, and Aristides beside Zerug. I pitied an acquaintance of mine for being named Peleg, till I found he had baptized his two boys Peleg and Seth. On a table in a little wayside inn I found Fox's Martyr's; and against the wall hung a framed sampler, with the following lines worked upon it:— Jesus, permit thine awful name to stand As the first offering of an infant's hand: And as her fingers o'er the canvass move, Oh fill her thoughtful bosom with thy love, With thy dear children let her bear a part, And write thy name thyself upon her heart. In these small inns the disagreeable practice of rocking in the chair is seen in its excess. In the inn parlour are three or four rocking- chairs, in which sit ladies who are vibrating in different directions, and at various velocities, so as to try the head of a stranger almost as severely as the tobacco-chewer his stomach. How this lazy and ungraceful indulgence ever became general, I cannot imagine; but the nation seems so wedded to it, that I see little chance of its being forsaken. When American ladies come to live in Europe, they
- 72. sometimes send home for a rocking-chair. A common wedding- present is a rocking-chair. A beloved pastor has every room in his house furnished with a rocking-chair by his grateful and devoted people. It is well that the gentlemen can be satisfied to sit still, or the world might be treated with the spectacle of the sublime American Senate in a new position; its fifty-two senators see- sawing in full deliberation, like the wise birds of a rookery in a breeze. If such a thing should ever happen, it will be time for them to leave off laughing at the Shaker worship. As we approached Greenbush, which lies opposite to Albany, on the east bank of the Hudson, we met riding horses, exercised by grooms, and more than one handsome carriage; tokens that we were approaching some centre of luxury. The view of Albany rising from the river side, with its brown stone courthouse and white marble capitol, is fine; but it wants the relief of more trees within itself, or of a rural background. How changed is this bustling city, thronged with costly buildings, from the Albany of the early days of Mrs. Grant of Laggan, when the children used to run up and down the green slope which is now State-street, imposing from its width and the massiveness of the houses seen behind its rows of trees! A tunnel is about to be made under the Hudson at Albany; meantime we crossed, as everybody does, by a horse ferryboat; a device so cruel as well as clumsy, that the sooner it is superseded the better. I was told that the strongest horses, however kept up with corn, rarely survive a year of this work. We observed that, even in this city, the physicians have not always their names engraved on brass doorplates. On the most conspicuous part of their houses, perhaps on the angle of a corner house, is nailed some glazed substance like floorcloth, with Dr. Such-an-one painted upon it. At Washington I remember seeing Magistrate thus affixed to a mere shed. As we surmounted the hill leading to our hotel, we saw our two shipmates dancing down the steps to welcome us. There certainly
- 73. is a feeling among shipmates which does not grow out of any other relation. They are thrown first into such absolute dependance on one another, for better for worse, and are afterward so suddenly and widely separated, that if they do chance to meet again, they renew their intimacy with a fervour which does not belong to a friendship otherwise originated. The glee of our whole party this evening is almost ridiculous to look back upon. Everything served to make a laugh, and we were almost intoxicated with the prospect of what we were going to see and do together. We had separated only a fortnight ago, but we had as much to talk over as if we had been travelling apart for six months. The Prussian had to tell his adventures, we our impressions, and the Southerner his comparisons of his own country with Europe. Then we had to arrange the division of labour by which the gentlemen were to lighten the cares of travelling. Dr. J., the Prussian, was on all occasions to select apartments for us; Mr. S., the Dutchman, to undertake the eating department; Mr. H., the American, was paymaster; and Mr. O., the German, took charge of the luggage. It was proposed that badges should be worn to designate their offices. Mr. S. was to be adorned with a corncob. Mr. H. stuck a bankbill in front of his hat; and, next morning, when Mr. O. was looking another way, the young men locked a small padlock upon his button-hole, which he was compelled to carry there for a day or two, till his comrades vouchsafed to release him from his badge. The hotel was well furnished and conducted. I pointed out, with some complacency, what a handsome piano we had in our drawing-room; but when, in the dark hour, I opened it in order to play, I found it empty of keys! a disappointment, however, which I have met with in England. Mr. Van Buren and his son happened to be in Albany, and called on me this afternoon. There is nothing remarkable in the appearance of this gentleman, whom I afterward saw frequently at Washington. He is small in person, with light hair and blue eyes. I
- 74. was often asked whether I did not think his manners gentlemanly. There is much friendliness in his manners, for he is a kind-hearted man; he is also rich in information, and lets it come out on subjects in which he cannot contrive to see any danger in speaking. But his manners want the frankness and confidence which are essential to good breeding. He questions closely, without giving anything in return. Moreover, he flatters to a degree which so cautious a man should long ago have found out to be disagreeable; and his flattery is not merely praise of the person he is speaking to, but a worse kind still; a skepticism and ridicule of objects and persons supposed to be distasteful to the one he is conversing with. I fully believe that he is an amiable and indulgent domestic man, and a reasonable political master, a good scholar, and a shrewd man of business; but he has the skepticism which marks the lower order of politicians. His public career exhibits no one exercise of that faith in men and preference of principle to petty expediency by which a statesman shows himself to be great. The consequence is, that, with all his opportunities, no great deed has ever been put to his account, and his shrewdness has been at fault in some of the most trying crises of his career. The man who so little trusts others, and so intensely regards self as to make it the study of his life not to commit himself, is liable to a more than ordinary danger of judging wrong when compelled, by the pressure of circumstances, to act a decided part. It has already been so with Mr. Van Buren more than once; and now that he is placed in a position where he must sometimes visibly lead, and cannot always appear to follow, it will be seen whether a due reverence of men and a forgetfulness of self would not have furnished him with more practical wisdom than all his sounding on his dim and perilous way. Mr. Calhoun is, I believe, Mr. Van Buren's evil genius. Mr. Calhoun was understood to be in expectation of succeeding to the presidential chair when Mr. Van Buren was appointed minister to Great Britain. This appointment of President Jackson's did not receive the necessary sanction from the Senate, and the new minister was recalled on the first possible
- 75. day, Mr. Calhoun being very active in bringing him back. Mr. Calhoun was not aware that he was recalling one who was to prove a successful rival. Mr. Calhoun has not been president; Mr. Van Buren is so; but the successful rival has a mortal dread of the great nullifier; a dread so obvious, and causing such a prostration of all principle and all dignity, as to oblige observers to conclude that there is more in the matter than they see; that it will come out some day why the disappointed aspirant is still to be propitiated, when he seems to be deprived of power to do mischief. In Society in America I have given an account of the nullification struggle, and of the irritation, the mysterious discontent which it has left behind. [2] Perhaps Mr. Van Buren may entertain the opinion which many hold, that that business is not over yet, and that the slavery question is made a pretext by the nullifiers of the South for a line of action to which they are impelled by the disappointed personal ambition of one or two, and the wounded pride of the many, who cannot endure the contrast between the increase of the free states of the North and the deterioration of the slave states of the South. However this may be, to propitiate Mr. Calhoun seems to have been Mr. Van Buren's great object for a long time past; an object probably hopeless in itself, and in the pursuit of which he is likely to lose the confidence of the North far faster than he could, at best, disarm the enmity of the South. In the spring of 1836, when Mr. Van Buren was still vice-president, and the presidential election was drawing near, Mr. Calhoun brought forward in the Senate his bill (commonly called the Gag Bill) to violate the postoffice function, by authorizing postmasters to investigate the contents of the mails, and to keep back all papers whatsoever relating to the subject of slavery. The bill was, by consent, read the first and second times without debate; and the Senate was to be divided on the question whether it should go to a third reading. The votes were equal, 18 to 18. Where's the
- 76. vice-president? shouted Mr. Calhoun's mighty voice. The vice- president was behind a pillar, talking. He was compelled to give the casting vote, to commit himself for once; a cruel necessity to a man of his caution. He voted for the third reading, and there was a bitter cry on the instant, The Northern States are sold. The bill was thrown out on the division on the third reading, and the vice- president lost by his vote the good-will of the whole body of abolitionists, who had till then supported him as the democratic and supposed anti-slavery candidate. As it was, most of the abolitionists did not vote at all, for want of a good candidate, and Mr. Van Buren's majority was so reduced as to justify a belief, that if the people had had another year to consider his conduct in, or if another democratic candidate could have been put forward, he would have been emphatically rejected. Having once committed himself, he has gone further still in propitiation of Mr. Calhoun. On the day of his presidential installation he declared that under no circumstances would he give his assent to any bill for the abolition of slavery in the District of Columbia. This declaration does not arise out of a belief that Congress has not power to abolish slavery in the District; for he did, not long before, when hard pressed, declare that he believed Congress to possess that power. He has therefore hazarded the extraordinary declaration that he will not, under any circumstances, assent to what may become the will of the people constitutionally imbodied. This is a bold intimation for a non-committal man to make. It remains to be seen whether Mr. Calhoun, if really dangerous, can be kept quiet by such fawning as this; and whether the will of the people may not be rather stimulated than restrained by this sacrifice of them to the South, so as either to compel the president to retract his declaration before his four years are out, or to prevent his re-election. How strange it is to recall one's first impressions of public men in the midst of one's matured opinions of them! How freshly I remember the chat about West Point and Stockbridge acquaintances that I had that afternoon at Albany, with the conspicuous man about whom I was then ignorant and indifferent,
- 77. and whom I have since seen committed to the lowest political principles and practices, while elected as professing some of the highest! It only remains to be said, that if Mr. Van Buren feels himself aggrieved by the interpretation which is commonly put upon the facts of his political life, he has no one to blame but himself; for such misinterpretation (if it exists) is owing to his singular reserve; a reserve which all men agree in considering incompatible with the simple honesty and cheerful admission of responsibility which democratic republicans have a right to require of their rulers. Before breakfast the next morning we walked down to the Padroon's house, known by reputation, with the history of the estate, to everybody. We just caught a sight of the shrubbery, and took leave to pass through the courtyard, and hastened back to breakfast, immediately after which we proceeded by railroad to Schenectady. There we at once stepped into a canalboat for Utica. I would never advise ladies to travel by canal, unless the boats are quite new and clean; or, at least, far better kept than any that I saw or heard of on this canal. On fine days it is pleasant enough sitting outside (except for having to duck under the bridges every quarter of an hour, under penalty of having one's head crushed to atoms), and in dark evenings the approach of the boatlights on the water is a pretty sight; but the horrors of night and of wet days more than compensate for all the advantages these vehicles can boast. The heat and noise, the known vicinity of a compressed crowd, lying packed like herrings in a barrel, the bumping against the sides of the locks, and the hissing of water therein like an inundation, startling one from sleep; these things are very disagreeable. We suffered under an additional annoyance in the presence of sixteen Presbyterian clergymen, some of the most unprepossessing of their class. If there be a duty more obvious than another on board a canalboat, it is to walk on the bank occasionally in fair weather, or, at least, to remain outside, in order to air the cabin (close enough at best) and get rid of the scents of the table before the unhappy passengers are shut up to sleep
- 78. there. These sixteen gentlemen, on their way to a Convention at Utica, could not wait till they got there to begin their devotional observances, but obtruded them upon the passengers in a most unjustifiable manner. They were not satisfied with saying an almost interminable grace before and after each meal, but shut up the cabin for prayers before dinner; for missionary conversation in the afternoon, and for scripture reading and prayers quite late into the night, keeping tired travellers from their rest, and every one from his fair allowance of fresh air. The passengers were all invited to listen to and to question a missionary from China who was of the party. The gentleman did not seem to have profited much by his travels, however; for he declared himself unable to answer some very simple inquiries. Is the religion of the Christian missionaries tolerated by the Chinese government? I am not prepared to answer that question. Are the Chinese cannibals? I am not prepared to answer that question. One requested that any brother would offer a suggestion as to how government might be awakened to the sinfulness of permitting Sunday mails; during the continuance of which practice there was no hope of the Sabbath being duly sanctified. No one was ready with a suggestion, but one offered a story, which every head was bent to hear. The story was of two sheep-drovers, one of whom feared God, and the other did not. The profane drover set out with his sheep for a particular destination two hours earlier than the other, and did not rest on Sunday like his pious comrade. What was the catastrophe? The Godfearing drover, though he had stood still all Sunday, arrived at his destination two hours earlier than the other. Ah! Ah! resounded through the cabin in all conceivable tones of conviction, no one asking particulars of what had happened on the road; of how and where the profane drover had been delayed. Temperance was, of course, a great topic with these divines, and they fairly provoked ridicule upon it. One passenger told me that they were so strict that they would not drink water out of the Brandywine
- 79. river; and another remarked that they partook with much relish of the strong wine-sauce served with our puddings. In addition to other discomforts, we passed the fine scenery of Little Falls in the night. I was not aware what we had missed till I traversed the Mohawk valley by a better conveyance nearly two years afterward. I have described this valley in my other work on America, [3] and must therefore restrain my pen from dwelling on its beauties here. The appearance of the berths in the ladies' cabin was so repulsive, that we were seriously contemplating sitting out all night, when it began to rain so as to leave us no choice. I was out early in the misty morning, however, and was presently joined by the rest of my party, all looking eagerly for signs of Utica being near. By eight o'clock we were at the wharf. We thought Utica the most extempore place we had yet seen. The right-up shops, the daubed houses, the streets running into the woods, all seemed to betoken that the place had sprung up out of some sudden need. How much more ancient and respectable did it seem after my return from the West, where I had seen towns so much newer still! We were civilly received and accommodated at Bagg's hotel, where we knew how to value cold water, spacious rooms, and retirement, after the annoyances of the boat. Our baggage-master was fortunate in securing a neat, clean stage to take us to Trenton Falls (14 miles), where we promised each other to spend the whole day, on condition of being off by five the next morning, in order to accomplish the distance to Syracuse in the course of the day. The reason for our economy of time was not merely that it was late in the season, and every day which kept us from the Falls of Niagara, therefore, of consequence, but that our German friend, Mr. O., was obliged to be back in New- York by a certain day. We all considered a little extra haste and fatigue a small tax to pay for the privilege of his companionship.
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