D0621619
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The document discusses I-functions and H-functions, which generalize Fox's H-function and contain certain Feynman integrals and partition functions as special cases. It establishes new double integral relations for products involving I-functions and H-functions. Specifically, it derives double integral relations for (A) a product of an I-function and H-function, and (B) the composition of an H-function with another H-function, under certain conditions on the parameters. These relations unify and extend previous known results for various special functions.
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 16-19
www.iosrjournals.org
www.iosrjournals.org 16 | Page
I- Function and H -Function Associated With Double Integral
Ashok Singh Shekhawat*
, Rakeshwar Purohit**
And Jyoti Shaktawat***
*Department of Mathematics, Regional college for Education, Research and Technology,
Jaipur, Rajasthan (India)
**Department of Mathematics and Statistics, University College of Science, Mohan Lal Sukhadia University,
Udaipur, Rajasthan (India)
***
Department of Mathematics, Kautilya Institute of Technology and Engineering,
Jaipur, Rajasthan (India)
Abstract: The object of this paper is to discuss certain integral properties of a I -function and H -function,
proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition of a Gaussian
model in Statistical Mechanics and several other functions as its particular cases. During the course of finding,
we establish certain new double integral relation pertaining to a product involving I function and H function.
These double integral relations are unified in nature and act as a key formulae from which we can obtain as their
special case, double integral relations concerning a large number of simple special functions. For the sake of
illustration, we record here some special cases of our main results which are also new and of interest by
themselves. All the result which are established in this paper are basic in nature and are likely to find useful
applications in several fields notably electrical network, probability theory and statistical mechanics.
Key words. I function, H -function, Hermite polynomials, Laguerre polynomials.
I. Introduction
The H -function [6] is a new generalization of the well known Fox’s H-function [4]. The H -function
pertains the exact partition function of the Gaussian model in statistical mechanics, functions useful in testing
hypothesis and several others as its particular cases. The conventional formulation may fail pertaining to the
domain of quantum cosmology but Feynman path integrals apply [10,11]. Feynman integral are useful in the
study and development of simple and multiple variable hypergeometric series which in turn are useful in
statistical mechanics.
The I-function defined as
jijiajja
jijibjj(b
zII[z]
zI ip1,njijian1,jja
iq1,mjijibm1,jj{(b
nm,
iqip
dszs)
2
1 s
where
s)as)b1
s)a1s)b
s)
jiji
ip
1nj
jiji
iq
1mj1i
jj
n
1j
jj
m
1j
…(1.1)
The H -function will be defined and represent as given in [1]
dx
i2
1
xHx]H
i
i
P1,NjjaN1,jAjj(a
Q1,MjjbM1,jj(b
NM,
QP,
NM,
QP,
…(1.2)
where](https://image.slidesharecdn.com/d0621619-130427010254-phpapp02/85/D0621619-1-320.jpg)
![I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 17 | Page
jj
P
1Nj
jj
Q
1Mj
jjj
N
1j
jj
M
1j
ab1
Aa1b
…(1.3)
which contains fractional powers of some of the gamma functions. Here aj (j = 1,…,P) and bj (j =
1,…,Q) are complex parameters, aj 0 (j = 1,…,P), j 0 (j = 1,…,Q) (not all zero simultaneously and the
exponents Aj (j = 1,…,N) and Bj (j = M+1,…,Q) can take on non-integer values. The contour in (1.2) is
imaginary axis R() = 0. It is suitably indented in order to avoid the singularities of the gamma functions and to
keep those singularities on appropriate side. Again for Aj (j=1,…,N) not an integer, the poles of the
gamma function of the numerator in (1.3) are converted to branch points. However, a long as there is no
coincidence of pole from any N)1,...,ja1andM)1,...,jb jjjj
pair, the
branch cuts can be chosen so that the path of integration can be distorted in the useful manner. For the sake of
brevity
0BAT j
P
1Nj
jj
Q
1Mj
jj
N
1j
j
M
1j
II. Main Result
We will obtain the following result:
(A) dydx
xy1
yv1
Hvy
xy1
x1
I
y)x)(1(1
xy1
xy1
y1
vy
xy1
x1 NM,
QP,
1
0
1
0
vHvIs
P1,NjjaN1,jAjja11
1k1Q1,MjBjjbM1,jjb
1NM,
1Q1,P
jijiajja
jijibjj(b
…(2.1)
provided that T
2
1
|Varg0b[R jj
Proof. We have
s)as)b1
sa1s)b
2
1
v
xy1
y)1
Hvy
xy1
x1
I
jiji
ip
1nj
jiji
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP,
.
dv
xy1
y1
ab1
Aa1b
i2
1
dsvy
xy1
x1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
s
…(2.2)
Multiplying both sides of (2.2) by
y)x)(1(1
xy1
xy1
y1
y
xy1
x1
and integration with respect to x and y between 0 and 1 for both the variable and making a use of a known result
[2, p.145],we get the required result (2.1) after a little simplification](https://image.slidesharecdn.com/d0621619-130427010254-phpapp02/85/D0621619-2-320.jpg)
![I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 18 | Page
(B) dvduv]HI(u)uvv)u
NM,
QP,
11
00
dzzz)zIs 1
0
jijiajja
jijibjj(b
. dzzH
P1,NjjaN1,jAjja11
1s1Q1,MjBjjbM1,jjb
1NM,
1Q1,P
…(2.3)
provided that 0bR( jj
Proof. Using (1.1) and (1.2), we have
dsu
s)as)b1
sa1s)b
2
1
v]HI(u) s
jiji
ip
1nj
jij
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP,
.
dv
ab1
Aa1b
i2
1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
…(2.4)
Multiplying both side by
11
uvv)u
and integrating with respect to u and v between 0 and
for both the variable and make a use of a known result [2, p.177], we get the required result. Letting
pz
ez)
in (2.3), we get the particular case after simplification
(c) dvduv]1Hu)]I[(v(1vv)1u)(1f(uv)
NM,
QP,
11
1
0
1
0
1s
1
0
jijiajja
jijibjj(b
z)(1f(z)z)(1Is
. dzz)1H
P1,NjjaN1,jAjja11
1s1Q1,MjBjjbM1,jjb
1NM,
1Q1,P
…(2.5)
provided that R() > 0, R() > 0.
Proof. Using equation (1.1) and (1.2), we have
dsu)1v
s)as)b1
sa1s)b
2
1
v]1Hu)]I[v(1 ss
jiji
ip
1nj
jij
iq
1mj1i
jj
n
1j
jj
m
1jNM,
QP,
](https://image.slidesharecdn.com/d0621619-130427010254-phpapp02/85/D0621619-3-320.jpg)
![I- Function And H -Function Associated With Double Integral
www.iosrjournals.org 19 | Page
.
dv)1
ab1
Aa1b
i2
1
jj
P
1Nj
jjj
Q
1Mj
jjj
N
1j
jj
M
1j
i
i
…(2.6)
Multiplying both side of (2.6) by
vv)1u)(1f(uv) 11
and integrating with respect to u and v
between 0 and 1 for both the variable and use of result [2, p.243] and by further simplification, we get the result
(2.5).
Letting f(z) =
1
z
in (2.5), we get the particular result after simplification.
Particular Case
(i) Taking
vy
xy1
x1
Svy
xy1
x1
I m
n
The result in (2.1), (2.2) and (2.3) reduces to the known result after a slight simplification obtained by Chaurasia
and Shekhawat [2].
Acknowledgement.
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and
valuable suggestions in the preparation of this paper.
References
[1]. R.G. Buschman and H.M. Srivastava, The H -function associated with a certain class of Feynman integrals, J. Phys. A: Math. Gen..
23 (1990), 4707-4710.
[2]. V.B.L. Chaurasia and Ashok Singh Shekhawat, Some integral properties of a general class of polynomials associated with Feynman
integrals, Bull. Malaysian Math. Sc. Soc. (2) (28) (2) (2005), 183-189.
[3]. J. Edewards, A Treatise on integral calculus, Chelsea Pub. Co., 2 (1922).
[4]. C. Fox, The G and H-functions as Symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429.
[5]. C. Grosche and F. Steiner, Hand Book of Feynman Path integrals, Springer Tracts in Modern Physics Vol.145, Springer-Verlag Berlin
Heidelberg, New York, 1998.
[6]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals : I. Transformation and reduction
formulae, J. Phys. A: Math. Gen. 20 (1987), 4109-4117.
[7]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals :II. A generalization of the H-
function, J. Phys. A: Math. Gen. 20 (1987), 4119-4128.
[8]. Saxena, R.K., On fractional integer operators, Math. Zeitschr, 96 (1967), 288-291.
[9]. Saxena, R.K. and Gupta, N., Some Abelian theorems for distributional H-function Transformation, Indian J. Pure Appl. Math. 25(8)
(1994), 869-879.
[10]. H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6.
[11]. H.M. Srivastava and N.P. Singh, The integration of certain products of Multivariable H-function with a general class of Polynomials,
Rend. Circ. Mat. Palermo 2(32) (1983), 157-187.
[12]. C. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 Fourth edition, Amer. Math. Soc. Providence, Rhode Island
(1975).](https://image.slidesharecdn.com/d0621619-130427010254-phpapp02/85/D0621619-4-320.jpg)
D0621619
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 16-19 www.iosrjournals.org www.iosrjournals.org 16 | Page I- Function and H -Function Associated With Double Integral Ashok Singh Shekhawat* , Rakeshwar Purohit** And Jyoti Shaktawat*** *Department of Mathematics, Regional college for Education, Research and Technology, Jaipur, Rajasthan (India) **Department of Mathematics and Statistics, University College of Science, Mohan Lal Sukhadia University, Udaipur, Rajasthan (India) *** Department of Mathematics, Kautilya Institute of Technology and Engineering, Jaipur, Rajasthan (India) Abstract: The object of this paper is to discuss certain integral properties of a I -function and H -function, proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition of a Gaussian model in Statistical Mechanics and several other functions as its particular cases. During the course of finding, we establish certain new double integral relation pertaining to a product involving I function and H function. These double integral relations are unified in nature and act as a key formulae from which we can obtain as their special case, double integral relations concerning a large number of simple special functions. For the sake of illustration, we record here some special cases of our main results which are also new and of interest by themselves. All the result which are established in this paper are basic in nature and are likely to find useful applications in several fields notably electrical network, probability theory and statistical mechanics. Key words. I function, H -function, Hermite polynomials, Laguerre polynomials. I. Introduction The H -function [6] is a new generalization of the well known Fox’s H-function [4]. The H -function pertains the exact partition function of the Gaussian model in statistical mechanics, functions useful in testing hypothesis and several others as its particular cases. The conventional formulation may fail pertaining to the domain of quantum cosmology but Feynman path integrals apply [10,11]. Feynman integral are useful in the study and development of simple and multiple variable hypergeometric series which in turn are useful in statistical mechanics. The I-function defined as jijiajja jijibjj(b zII[z] zI ip1,njijian1,jja iq1,mjijibm1,jj{(b nm, iqip dszs) 2 1 s where s)as)b1 s)a1s)b s) jiji ip 1nj jiji iq 1mj1i jj n 1j jj m 1j …(1.1) The H -function will be defined and represent as given in [1] dx i2 1 xHx]H i i P1,NjjaN1,jAjj(a Q1,MjjbM1,jj(b NM, QP, NM, QP, …(1.2) where
- 2. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 17 | Page jj P 1Nj jj Q 1Mj jjj N 1j jj M 1j ab1 Aa1b …(1.3) which contains fractional powers of some of the gamma functions. Here aj (j = 1,…,P) and bj (j = 1,…,Q) are complex parameters, aj 0 (j = 1,…,P), j 0 (j = 1,…,Q) (not all zero simultaneously and the exponents Aj (j = 1,…,N) and Bj (j = M+1,…,Q) can take on non-integer values. The contour in (1.2) is imaginary axis R() = 0. It is suitably indented in order to avoid the singularities of the gamma functions and to keep those singularities on appropriate side. Again for Aj (j=1,…,N) not an integer, the poles of the gamma function of the numerator in (1.3) are converted to branch points. However, a long as there is no coincidence of pole from any N)1,...,ja1andM)1,...,jb jjjj pair, the branch cuts can be chosen so that the path of integration can be distorted in the useful manner. For the sake of brevity 0BAT j P 1Nj jj Q 1Mj jj N 1j j M 1j II. Main Result We will obtain the following result: (A) dydx xy1 yv1 Hvy xy1 x1 I y)x)(1(1 xy1 xy1 y1 vy xy1 x1 NM, QP, 1 0 1 0 vHvIs P1,NjjaN1,jAjja11 1k1Q1,MjBjjbM1,jjb 1NM, 1Q1,P jijiajja jijibjj(b …(2.1) provided that T 2 1 |Varg0b[R jj Proof. We have s)as)b1 sa1s)b 2 1 v xy1 y)1 Hvy xy1 x1 I jiji ip 1nj jiji iq 1mj1i jj n 1j jj m 1jNM, QP, . dv xy1 y1 ab1 Aa1b i2 1 dsvy xy1 x1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i s …(2.2) Multiplying both sides of (2.2) by y)x)(1(1 xy1 xy1 y1 y xy1 x1 and integration with respect to x and y between 0 and 1 for both the variable and making a use of a known result [2, p.145],we get the required result (2.1) after a little simplification
- 3. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 18 | Page (B) dvduv]HI(u)uvv)u NM, QP, 11 00 dzzz)zIs 1 0 jijiajja jijibjj(b . dzzH P1,NjjaN1,jAjja11 1s1Q1,MjBjjbM1,jjb 1NM, 1Q1,P …(2.3) provided that 0bR( jj Proof. Using (1.1) and (1.2), we have dsu s)as)b1 sa1s)b 2 1 v]HI(u) s jiji ip 1nj jij iq 1mj1i jj n 1j jj m 1jNM, QP, . dv ab1 Aa1b i2 1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i …(2.4) Multiplying both side by 11 uvv)u and integrating with respect to u and v between 0 and for both the variable and make a use of a known result [2, p.177], we get the required result. Letting pz ez) in (2.3), we get the particular case after simplification (c) dvduv]1Hu)]I[(v(1vv)1u)(1f(uv) NM, QP, 11 1 0 1 0 1s 1 0 jijiajja jijibjj(b z)(1f(z)z)(1Is . dzz)1H P1,NjjaN1,jAjja11 1s1Q1,MjBjjbM1,jjb 1NM, 1Q1,P …(2.5) provided that R() > 0, R() > 0. Proof. Using equation (1.1) and (1.2), we have dsu)1v s)as)b1 sa1s)b 2 1 v]1Hu)]I[v(1 ss jiji ip 1nj jij iq 1mj1i jj n 1j jj m 1jNM, QP,
- 4. I- Function And H -Function Associated With Double Integral www.iosrjournals.org 19 | Page . dv)1 ab1 Aa1b i2 1 jj P 1Nj jjj Q 1Mj jjj N 1j jj M 1j i i …(2.6) Multiplying both side of (2.6) by vv)1u)(1f(uv) 11 and integrating with respect to u and v between 0 and 1 for both the variable and use of result [2, p.243] and by further simplification, we get the result (2.5). Letting f(z) = 1 z in (2.5), we get the particular result after simplification. Particular Case (i) Taking vy xy1 x1 Svy xy1 x1 I m n The result in (2.1), (2.2) and (2.3) reduces to the known result after a slight simplification obtained by Chaurasia and Shekhawat [2]. Acknowledgement. The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valuable suggestions in the preparation of this paper. References [1]. R.G. Buschman and H.M. Srivastava, The H -function associated with a certain class of Feynman integrals, J. Phys. A: Math. Gen.. 23 (1990), 4707-4710. [2]. V.B.L. Chaurasia and Ashok Singh Shekhawat, Some integral properties of a general class of polynomials associated with Feynman integrals, Bull. Malaysian Math. Sc. Soc. (2) (28) (2) (2005), 183-189. [3]. J. Edewards, A Treatise on integral calculus, Chelsea Pub. Co., 2 (1922). [4]. C. Fox, The G and H-functions as Symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429. [5]. C. Grosche and F. Steiner, Hand Book of Feynman Path integrals, Springer Tracts in Modern Physics Vol.145, Springer-Verlag Berlin Heidelberg, New York, 1998. [6]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals : I. Transformation and reduction formulae, J. Phys. A: Math. Gen. 20 (1987), 4109-4117. [7]. A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals :II. A generalization of the H- function, J. Phys. A: Math. Gen. 20 (1987), 4119-4128. [8]. Saxena, R.K., On fractional integer operators, Math. Zeitschr, 96 (1967), 288-291. [9]. Saxena, R.K. and Gupta, N., Some Abelian theorems for distributional H-function Transformation, Indian J. Pure Appl. Math. 25(8) (1994), 869-879. [10]. H.M. Srivastava, A contour integral involving Fox’s H-function, Indian J. Math. 14 (1972), 1-6. [11]. H.M. Srivastava and N.P. Singh, The integration of certain products of Multivariable H-function with a general class of Polynomials, Rend. Circ. Mat. Palermo 2(32) (1983), 157-187. [12]. C. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 Fourth edition, Amer. Math. Soc. Providence, Rhode Island (1975).