A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel-Maitland Function
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This document presents a study on various classes of polynomials linked to differential operators and generalizations of the Bessel-Maitland function. The authors utilize operational techniques to derive linear and bilateral generating relations, including those associated with the Stirling numbers of the second kind. Several definitions of polynomials and generating functions are detailed, along with specific cases and properties related to these mathematical structures.
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. VI (Jan - Feb. 2015), PP 55-63
www.iosrjournals.org
DOI: 10.9790/5728-11165563 www.iosrjournals.org 55 | Page
A Class of Polynomials Associated with Differential Operator and
with a Generalization of Bessel-Maitland Function
Manoj Singh1
, Mumtaz Ahmad Khan2
and Abdul Hakim Khan3
1
(Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia)
2,3
(Department of Applied Mahematics, Faculty of Engineering, Aligarh Muslim University, U.P., India)
Abstract:The object of this paper is to present several classes of linear and bilateral generating relations by
employing operational techniques, which reduces as a special case of (known or new) bilateral generating
relations.At last some of generating functions associated with stirling number of second kind are also discussed.
Mathematics Subject Classification(2010): Primary 42C05, Secondary 33C45.
Keywords: Generating relations, Differential operator, Rodrigue formula, Stirling number.
I. Introduction
In literature several authors have discussed a number of polynomials defined by their Rodrigue's
formula and gave several classes of linear, bilinear, bilateral and mixed multilateral generating functions.
In 1971, H.M. Srivastava and J.P.Singhal [1], introduced a general class of polynomial 𝐺𝑛
(𝛼)
(𝑥, 𝑟, 𝑝, 𝑘)
(of degree 𝑛 in 𝑥 𝑟
and in 𝛼) as
𝐺𝑛
𝛼
(𝑥, 𝑟, 𝑝, 𝑘) =
𝑥−𝛼−𝑘𝑛
𝑛!
exp(𝑝𝑥 𝑟
)(𝑥 𝑘+1
𝐷) 𝑛
{𝑥 𝛼
exp(−𝑝𝑥 𝑟
)} (1.1)
where Laguerre, Hermite, and Konhauser plynomials are the special case of (1.1).
In 2001, H.M. Srivastava [2], defined a polynomial set of degree 𝑛 in 𝑥 𝑟
, 𝛼 and 𝜂 by
𝜏 𝑛
𝛼
(𝑥; 𝑟, 𝛽, 𝑘, 𝜂) =
𝑥−𝛼−𝑘𝑛
𝑛!
exp(𝛽𝑥 𝑟
){𝑥 𝑘
𝜂 + 𝑥𝐷 } 𝑛
𝑥 𝛼
exp −𝛽𝑥 𝑟
, 𝑛 ∈ 𝑁0 (1.2)
A detailed account of (1.2) is present in the work of Chen et al. [3], who derived several new classes of linear,
bilinear and mixed multilateral generating relations.
Recently in 2008, Shukla and Prajapati [4], introduced a class of polynomial which is connected by the
generalized Mittag-Leffler function 𝐸 𝛼,𝛽
𝛾,𝑞
(𝑧), whose properties were discussed in his paper [5], is defined as
𝐴 𝑞𝑛
𝛼,𝛽,𝛾,𝑞
(𝑥; 𝑎, 𝑘, 𝑠) =
𝑥−𝛿−𝑎𝑛
𝑛!
𝐸 𝛼,𝛽
𝛾,𝑞
{𝑝 𝑘 (𝑥)}{𝑥 𝑎
𝑠 + 𝑥𝐷 } 𝑛
𝑥 𝛿
𝐸 𝛼,𝛽
𝛾,𝑞
−𝑝 𝑘 𝑥 (1.3)
where 𝛼, 𝛽, 𝛾, 𝛿 are real or complex numbers and 𝑎, 𝑘, 𝑠 are constants.
In the present paper we have defined a polynomial by using the differential operator 𝜃, which involves
two parameters 𝑘 and 𝜆 independent of 𝑥 as
𝜃 = 𝑥 𝑘
𝜆 + 𝑥𝐷𝑥 , 𝐷𝑥 =
𝑑
𝑑𝑥
(1.4)
In connection with the differential operator (1.4), we define the polynomial 𝑀𝑞𝑛
(𝜇,𝑛𝑢,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) of degree 𝑛
in 𝑥 𝑟
, 𝜉 and 𝜆 as
𝑀𝑞𝑛
𝜇,𝜈,𝛾,𝜉
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) =
𝑥−𝜉−𝑘𝑛
𝑛!
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)𝜃 𝑛
{𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
(𝑝𝑥 𝑟
)} (1.5)
In (1.5), 𝜇, 𝜈, 𝛾, 𝜉 are real or complex numbers and 𝐽𝜈,𝑞
𝜇,𝛾
(𝑧) is the generalized Bessel's Maitland function for
𝜇, 𝜈, 𝛾 ∈ 𝐶; 𝑅𝑒(𝜇) ≥ 0, 𝑅𝑒(𝜈) ≥ 0, 𝑅𝑒(𝛾) ≥ 0 and 𝑞 ∈ (0,1) ∪ 𝑁 as
𝐽𝜈,𝑞
𝜇,𝛾
(𝑧) =
∞
𝑛=0
(𝛾) 𝑞𝑛 (−𝑧) 𝑛
𝑛! Γ 𝜇𝑛 + 𝜈 + 1
(1.6)
where (𝛾) 𝑞𝑛 =
Γ(𝛾+𝑞𝑛)
Γ(𝛾)
denotes the generalized Pochhammer symbol, which reduces to 𝑞 𝑞𝑛
𝑞
𝑟=1
𝛾+𝑟−1
𝑞 𝑛
, if
𝑞 ∈ 𝑁. Certain properties of (1.6) have been discussed in [6].
Some special cases of the polynomial (1.5) are given below:
𝑀 𝑛
1,0,1,𝛼
(𝑥; 𝑟, 𝑝, 𝑘, 0) = 𝐺𝑛
𝛼
(𝑥, 𝑟, 𝑝, 𝑘) (1.7)
where 𝐺𝑛
(𝛼)
(𝑥, 𝑟, 𝑝, 𝑘) is defined by equation (1.1).
𝑀 𝑛
1,0,1,𝛼
(𝑥; 𝑟, 𝛽, 𝑘, 𝜂) = 𝜏 𝑛
𝛼
(𝑥; 𝑟, 𝛽, 𝑘, 𝜂) (1.8)
where𝜏 𝑛
(𝛼)
(𝑥; 𝑟, 𝛽, 𝑘, 𝜂) is defined by equation (1.2).](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-1-320.jpg)
![A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel …
DOI: 10.9790/5728-11165563 www.iosrjournals.org 56 | Page
𝑀𝑞𝑛
𝛼,𝛽−1,𝛾,𝛿
(𝑥; 𝑘, 𝑝, 𝑎, 𝑠) = 𝐴 𝑞𝑛
𝛼,𝛽,𝛾,𝑞
(𝑥; 𝑎, 𝑘, 𝑠) (1.9)
where 𝐴 𝑞𝑛
(𝛼,𝛽,𝛾,𝑞)
(𝑥; 𝑎, 𝑘, 𝑠) is defined by equation (1.3).
II. Linear Generating Functions
By appealing the property of the operator of Patil and Thakre [7],
𝑒 𝑡𝜃
{𝑥 𝛼
𝑓(𝑥)} =
𝑥 𝛼
(1 − 𝑡𝑘𝑥 𝑘 )
𝛼+𝜆
𝑘
. 𝑓
𝑥
(1 − 𝑡𝑘𝑥 𝑘 )
1
𝑘
(2.1)
and
𝑒 𝑡𝜃
{𝑥 𝛼−𝑛
𝑓(𝑥)} = 𝑥 𝛼
(1 + 𝑘𝑡)−1+
𝛼+𝜆
𝑘 𝑓 𝑥(1 + 𝑘𝑡)
1
𝑘 (2.2)
one obtains certain generating relations for the polynomials defined by (1.5) as given below:
∞
𝑛=0
𝑀𝑞𝑛
𝜇,𝜈,𝛾,𝜉
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 − 𝑘𝑡)−
𝜉+𝜆
𝑘 𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
𝑝 𝑥 1 − 𝑘𝑡 −
1
𝑘
𝑟
(2.3)
(|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
∞
𝑛=0
𝑀𝑞𝑛
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝜉+𝜆
𝑘 𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)𝐽𝜈,𝑞
𝜇,𝛾
[𝑝{𝑥 1 + 𝑘𝑡)
1
𝑘 } 𝑟
(2.4)
(|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝑀𝑞(𝑚+𝑛)
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑡)−
1
𝑘 } 𝑟]
𝑀𝑞𝑚
𝜇,𝜈,𝛾,𝜉
(𝑥 1 − 𝑘𝑡)−
1
𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆 (2.5)
(𝑛 ∈ 𝑁0;|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝑀𝑞(𝑚+𝑛)
(𝜇,𝜈,𝛾,𝜉−𝑘𝑛)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 + 𝑘𝑡)−1+(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 + 𝑘𝑡)1/𝑘 } 𝑟]
𝑀𝑞𝑚
(𝜇,𝜈,𝛾,𝜉)
(𝑥(1 + 𝑘𝑡)1/𝑘
; 𝑟, 𝑝, 𝑘, 𝜆) (2.6)
(𝑛 ∈ 𝑁0;|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
Proof of (2.5): From (1.5), we write
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝑀𝑞 𝑚+𝑛
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛
=
∞
𝑛=0
𝑥−𝜉− 𝑚+𝑛 𝑘
𝑚! 𝑛!
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝜃 𝑚+𝑛
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
𝑝𝑥 𝑟
𝑡 𝑛
=
𝑥−𝜉−𝑚𝑘
𝑚!
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
∞
𝑛=0
𝑥−𝑛𝑘
𝑡 𝑛
𝜃 𝑛
𝑛!
. 𝜃 𝑚
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
𝑝𝑥 𝑟
=
𝑥−𝜉−𝑚𝑘
𝑚!
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
exp 𝑥−𝑘
𝑡𝜃 𝜃 𝑚
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
𝑝𝑥 𝑟
Again by using (1.5) and (2.1), we obtain the generating relation (2.5).
Proof of (2.6): Multiplying equation (2.4) by
𝑥 𝜉
𝐽 𝜈,𝑞
𝜇 ,𝛾
(−𝑝𝑥 𝑟)
and then operating upon both sides by the differential
operator 𝜃 𝑚
, we get
∞
𝑛=0
𝜃 𝑚
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟 )
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉−𝑘𝑛)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) 𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝜉+𝜆
𝑘 𝜃 𝑚
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
[𝑝{𝑥(1 + 𝑘𝑡)
1
𝑘 } 𝑟
] (2.7)
Now replacing 𝑛 by 𝑚 in (1.5), we obtain
𝑚!
𝑥 𝜉+𝑚𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝑀𝑞𝑚
𝜇,𝜈,𝛾,𝜉
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝜃 𝑚
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
𝑝𝑥 𝑟
(2.8)
Again replacing 𝑚 by 𝑚 + 𝑛 in equation (2.8), we obtain
(𝑚 + 𝑛)!
𝑥 𝜉+𝑚𝑘 +𝑛𝑘
𝐽 𝜈,𝑞
𝜇 ,𝛾
−𝑝𝑥 𝑟 𝑀𝑞 𝑚+𝑛
𝜇,𝜈,𝛾,𝜉
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝜃 𝑚+𝑛
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
𝑝𝑥 𝑟](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-2-320.jpg)
![A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel …
DOI: 10.9790/5728-11165563 www.iosrjournals.org 57 | Page
= 𝜃 𝑚
𝜃 𝑛
{𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
(𝑝𝑥 𝑟
)} = 𝑛!
𝑥 𝜉+𝑛𝑘
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟 )
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) (2.9)
Further replacing 𝜉 by (𝜉 − 𝑘𝑛) in (2.9), we obtain
𝑥 𝜉
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟)
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉−𝑘𝑛)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) =
𝑚 + 𝑛!
𝑛!
𝑥 𝜉+𝑚𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝑀𝑞 𝑚+𝑛
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 (2.10)
Substituting the value of equation (2.10) in equation (2.7) and using equation (2.8), we obtain
∞
𝑛=0
𝑚 + 𝑛!
𝑛!
𝑥 𝜉+𝑘𝑛
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝑀𝑞 𝑚+𝑛
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝜉+𝜆
𝑘 𝑚!
𝑥 𝜉+𝑘𝑛
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 + 𝑘𝑡)
1
𝑘 } 𝑟]
𝑀𝑞𝑚
𝜇,𝜈,𝛾,𝜉
x 1 + 𝑘𝑧
1
𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆
which gives the generating relation (2.6).
The theorem by H.M. Srivastava on mixed generating functions (cf. [8], p. 378, Th. 12), plays an
important role to derive several other generating relation for the polynomial 𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) defined by
(1.5).
Theorem 2.1:Let each of the functions 𝐴(𝑧),𝐵𝑗 (𝑧)(𝑗 = 1, . . . , 𝑚), and 𝑧−1
𝐶𝑙(𝑧)(𝑙 = 1, . . . , 𝑠) be analytic in a
neighborhood of the origin, and assume that
𝐴 0 . 𝐵 0 . 𝐶′
𝑙 0 ≠ 0 𝑗 = 1, . . . , 𝑚; 𝑙 = 1, . . . , 𝑠 . (2.11)
Define the sequence of function
𝑔 𝑛
(𝛼1,...,𝛼 𝑚 )
(𝑥1, . . . , 𝑥 𝑠)
𝑛=0
∞
by
𝐴(𝑧)
𝑚
𝑗 =1
{[𝐵𝑗 (𝑧)] 𝛼 𝑗 }. exp
𝑠
𝑙=1
𝑥𝑙 𝐶𝑙(𝑧) =
∞
𝑛=0
𝑔 𝑛
𝛼1,…,𝛼 𝑚
𝑥1, . . . , 𝑥 𝑠
𝑧 𝑛
𝑛!
(2.12)
where 𝛼1, . . . , , 𝛼 𝑚 and 𝑥1, . . . , 𝑥 𝑠 are arbitrary complex numbers independent of 𝑧.
Then, for arbitrary parameters 𝜆1, . . . , 𝜆 𝑚 and 𝑦1,. . . , 𝑦𝑠 independent of z,
∞
𝑛=0
𝑔 𝑛
(𝛼1+𝜆1 𝑛,...,𝛼 𝑚 +𝜆 𝑚 𝑛)
(𝑥1 + 𝑛𝑦1, . . . , 𝑥 𝑠 + 𝑛𝑦𝑠)
𝑡 𝑛
𝑛!
=
𝐴(𝜁) 𝑚
𝑗=1 {[𝐵𝑗 (𝜁)] 𝛼 𝑗 }. exp 𝑠
𝑙=1 𝑥𝑙 𝐶𝑙(𝜁)
1 − 𝜁 𝑚
𝑗=1 𝜆𝑗 [𝐵′𝑗 (𝜁)/𝐵𝑗 (𝜁)] + 𝑠
𝑙=1 𝑦𝑙 𝐶′𝑙(𝜁)
𝜁 = 𝑡
𝑚
𝑗 =1
{[𝐵𝑗 (𝜁)] 𝜆 𝑗 }exp
𝑠
𝑙=1
𝑦𝑙 𝐶𝑙(𝜁) ; 𝑚, 𝑠 ∈ 𝑁 (2.13)
By putting 𝜇 = 1, 𝜈 = 0, 𝛾 = 1 and 𝑞 = 1 in the generating relation (2.3), reduces into the form
∞
𝑛=0
𝑀 𝑛
1,0,1,𝜉
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 − 𝑘𝑡)−
𝜉+𝜆
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 − 𝑘𝑡)−
𝑟
𝑘 } (2.14)
The generating function (2.14) is equivalently of the type (2.12), with of course,
𝑚 − 1 = 𝑠 = 1, 𝐴(𝑧) = 1, 𝐵1(𝑧) = 𝐵2(𝑧) = (1 − 𝑘𝑡)−1/𝑘
, 𝑥1 = 𝑝, 𝐶1(𝑧) = 𝑝𝑥 𝑟
[1 − (1 − 𝑘𝑡)−
𝑟
𝑘 and
𝑔 𝑛
𝜉,𝜆
𝑝 = 𝑛! 𝑀 𝑛
1,0,1,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 ;(𝑛 ∈ 𝑁).
Thus applying the above theorem to the generating function (2.14), one obtains
∞
𝑛=0
𝑀 𝑛
(1,0,1,𝜉+𝛿𝑛)
(𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛
=
(1 − 𝑘𝜁)−(𝜉+𝜆)/𝑘
exp 𝑝𝑥 𝑟
{1 − (1 − 𝑘𝜁)−𝑟/𝑘
}
1 − 𝜁(1 − 𝑘𝜁)−1 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟(1 − 𝑘𝜁)−𝑟/𝑘
(2.15)
where,
𝜁 = 𝑡(1 − 𝑘𝜁)−(𝛿+𝑣)/𝑘
exp 𝑢𝑥 𝑟
{1 − (1 − 𝑘𝜁)−𝑟/𝑘
} ; 𝑘 ≠ 0
By setting 𝜁 by
𝜁
𝑘
(𝑘 ≠ 0), in the generating relation (2.15), one obtains
∞
𝑛=0
𝑀 𝑛
(1,0,1,𝜉+𝛿𝑛)
(𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛
=
(1 − 𝜁)−(𝜉+𝜆)/𝑘
exp 𝑝𝑥 𝑟
{1 − (1 − 𝜁)−𝑟/𝑘
}
1 − 𝑘−1 𝜁(1 − 𝜁)−1 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟 (1 − 𝜁)−𝑟/𝑘
(2.16)
where,
𝜁 = 𝑘𝑡(1 − 𝜁)−(𝛿+𝑣)/𝑘
exp 𝑢𝑥 𝑟
{1 − (1 − 𝜁)−𝑟/𝑘
} ; 𝑘 ≠ 0
By replacing 𝜁 by
𝜁
1+𝜁
in the generating function (2.16), one immediately obtains the generating function](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-3-320.jpg)
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DOI: 10.9790/5728-11165563 www.iosrjournals.org 58 | Page
∞
𝑛=0
𝑀 𝑛
(1,0,1,𝜉+𝛿𝑛)
𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛 𝑡 𝑛
=
(1 + 𝜁)(𝜉+𝜆)/𝑘
exp 𝑝𝑥 𝑟
{1 − (1 + 𝜁) 𝑟/𝑘
}
1 − 𝑘−1 𝜁 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟(1 + 𝜁) 𝑟/𝑘
(2.17)
where,
𝜁 = 𝑘𝑡(1 + 𝜁)1+(𝛿+𝑣)/𝑘
exp 𝑢𝑥 𝑟
{1 − (1 + 𝜁) 𝑟/𝑘
} ; 𝑘 ≠ 0
Again by setting 𝛿 = −𝑣 − 𝑘 and 𝑢 = 0 in (2.17), one obtains
∞
𝑛=0
𝑀 𝑛
(1,0,1,𝜉−(𝑣+𝑘)𝑛)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝜉+𝜆
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 + 𝑘𝑡)
𝑟
𝑘 } (2.18)
(|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
when, 𝑣 = 0, (2.18) readily gives us
∞
𝑛=0
𝑀 𝑛
(1,0,1,𝜉−𝑘𝑛 )
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝜉+𝜆
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 + 𝑘𝑡)
𝑟
𝑘 } (2.19)
(|𝑡| < |𝑘|−1
; 𝑘 ≠ 0)
The generating relation obtained (2.19) is the particular case of (2.4) at 𝜇 = 1, 𝜈 = 0, 𝛾 = 1 and 𝑞 = 1.
With the help of (1.8), the generating relation (2.15), (2.16), (2.17), (2.18), and (2.19) reduce to the
generating function (2.16), (2.19), (2.22), (2.24) and (2.51) of Chen et al. [3].
III. Bilateral Generating Functions
A class of function {𝑆𝑛 (𝑥), 𝑛 = 0,1,2, . . . } (see [8], p. 411) generated by
∞
𝑛=0
𝐴 𝑚,𝑛 𝑆 𝑚+𝑛 𝑡 𝑛
= 𝑓(𝑥, 𝑡){𝑔(𝑥, 𝑡)}−𝑚
𝑆 𝑚 ((𝑥, 𝑡)) (3.1)
where 𝑚 ≥ 0 is an integer, the coefficient 𝐴 𝑚,𝑛 are arbitrary constant and 𝑓, 𝑔, are suitable functions of 𝑥 and
𝑡.
Numerous classes of bilinear (or bilateral) generating functions were obtained for the function 𝑆𝑛 (𝑥) generated
by (3.1).
Theorem 3.1: (cf. [8], p. 412, Th. 13) For the sequence {𝑆𝑛 (𝑥)} generated by (3.1), let
𝐹 𝑥, 𝑡 =
∞
𝑛=0
𝑎 𝑛 𝑆𝑛 𝑥 𝑡 𝑛
(3.2)
where 𝑎 𝑛 ≠ 0 are arbitrary constant,
Then,
𝑓 𝑥, 𝑡 𝐹 𝑥, 𝑡 ,
𝑦𝑡
𝑔 𝑥, 𝑡
=
∞
𝑛=0
𝑆𝑛 𝑥 𝜎𝑛 𝑦 𝑡 𝑛
(3.3)
where 𝜎𝑛 (𝑦) is a polynomial (of degree 𝑛 in 𝑦) defined by
𝜎𝑛 𝑦 =
𝑛
𝑘=0
𝑎 𝑘 𝐴 𝑘,𝑛−𝑘 𝑦 𝑘
(3.4)
By using the theorem 3.1, we obtain the bilateral generating function for the polynomial (1.5) as
∞
𝑛=0
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝜎𝑛 (𝑦)𝑡 𝑛
= (1 − 𝑘𝑡)−
𝜉+𝜆
𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑡)−
1
𝑘 } 𝑟]
𝐹 𝑥(1 − 𝑘𝑡)−
1
𝑘 , 𝑦𝑡(1 − 𝑘𝑡)−1
(3.5)
where,
𝐹(𝑥, 𝑡) =
∞
𝑛=0
(𝑎) 𝑛 𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
Proof of (3.5):With suitable replacement in theorem 3.1, we obtain
∞
𝑛=0
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝜎𝑛 (𝑦)𝑡 𝑛
=
∞
𝑛=0
𝑀𝑞𝑛
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)
𝑛
𝑚=0
𝑛
𝑚
(𝑎) 𝑚 (𝑦) 𝑚
𝑡 𝑛](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-4-320.jpg)
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=
∞
𝑚=0
(𝑎) 𝑚 (𝑦𝑡) 𝑚
∞
𝑛=0
𝑚 + 𝑛
𝑚
𝑀𝑞(𝑚+𝑛)
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛
Making use of generating relation (2.5), yields the bilateral generating function (3.5).
Particular cases of 3.5:
(i) If we put 𝜇 = 1, 𝜈 = 0, 𝛾 = 1, 𝜆 = 0, 𝑞 = 1, 𝜉 = 𝛼 and 𝑎 = 𝜇 in (3.5), which in conjunction with the
generating function (cf. [1]),
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝐺 𝑚+𝑛
(𝛼)
(𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛
= (1 − 𝑘𝑡)−𝑚−
𝛼
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 − 𝑘𝑡)−
𝑟
𝑘 } 𝐺 𝑚
𝛼
𝑥(1 − 𝑘𝑡)−
1
𝑘 , 𝑟, 𝑝, 𝑘 (3.6)
and the use of (1.7), we get bilinear generating function obtained by Srivastava and Singhal [1],
∞
𝑛=0
𝑛! 𝐺𝑛
𝛼
(𝑥, 𝑟, 𝑝, 𝑘)𝜎𝑛 (𝑦)𝑡 𝑛
= (1 − 𝑘𝑡)−
𝛼
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 − 𝑘𝑡)−
𝑟
𝑘 } 𝐹 𝑥(1 − 𝑘𝑡)−
1
𝑘 , 𝑦𝑡(1 − 𝑘𝑡)−1
(3.7)
where,
𝐹(𝑥, 𝑡) =
∞
𝑛=0
(𝜇) 𝑛 𝐺𝑛
(𝛼)
(𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛
(ii) Again, putting 𝜇 = 𝛼, 𝜈 = 𝛽 − 1, 𝑝𝑥 𝑟
= 𝑝 𝑘 (𝑥), 𝜉 = 𝛿, 𝑘 = 𝑎, 𝜆 = 𝑠 and 𝑎 = 𝜇 in (3.5), which in conjunction
with the generating function (cf. [4]; p. 26. Eq. (1.6))
∞
𝑚=0
𝑚 + 𝑛
𝑛
𝐴 𝑞(𝑚+𝑛)
(𝛼,𝛽,𝛾,𝛿)
(𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑚
= (1 − 𝑎𝑡)−𝑛−
𝛿+𝑠
𝑎
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 𝑥
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 {𝑥(1 − 𝑎𝑡)−
1
𝑎 }
𝐴 𝑞𝑛
𝛼,𝛽,𝛾,𝛿
𝑥(1 − 𝑎𝑡)−
1
𝑎 ; 𝑎, 𝑘, 𝑠 (3.8)
and use of (1.9), we obtain a new bilateral generating function for the polynomial defined by Shukla and
Prajapati [4] as
∞
𝑛=0
𝐴 𝑞𝑛
(𝛼,𝛽,𝛾,𝛿)
(𝑥; 𝑎, 𝑘, 𝑠)𝜎𝑛 (𝑦)𝑡 𝑛
= (1 − 𝑎𝑡)−
𝛿+𝑠
𝑎
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 𝑥
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 {𝑥(1 − 𝑎𝑡)−
1
𝑎 }
𝐹 𝑥(1 − 𝑎𝑡)−
1
𝑎 , 𝑦𝑡(1 − 𝑎𝑡)−1
(3.9)
where,
𝐹(𝑥, 𝑡) =
∞
𝑛=0
(𝜇) 𝑛 𝐴 𝑞𝑛
(𝛼,𝛽,𝛾,𝛿)
(𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑛
In 1990, Hubble and Srivastava [9] generalize the theorem.
Theorem 3.2:Corresponding to the function 𝑆𝑛 (𝑥), generated by (3.1), let
Θ 𝑁 𝑥, 𝑦, 𝑡 =
∞
𝑛=0
𝑎 𝑛 𝑆𝑛 𝑥 𝐿 𝑁
𝜆+𝑛
𝑦 𝑡 𝑛
, 𝑎 𝑛 ≠ 0 (3.10)
where 𝜆 is an arbitrary (real or complex) parameter. Suppose also that
𝜃 𝑚,𝑛 𝑧, 𝜔, 𝑥 =
𝑚𝑖𝑛 {𝑚,𝑛}
𝑘=0
(−1) 𝑘
𝑘
𝑎 𝑚−𝑘 𝐴 𝑚−𝑘,𝑛−𝑘 𝑧 𝑚−𝑘
𝜔 𝑛−𝑘
𝑆 𝑚+𝑛−2𝑘 𝑥 (3.11)
Then,
∞
𝑚,𝑛=0
𝜃 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
𝜆+𝑚
(𝑦)𝑡 𝑚
= exp −𝑡 𝑓 𝑥, 𝜔 Θ 𝑁 𝑥, 𝜔 , 𝑦 + 𝑡,
𝑧𝑡
𝑔 𝑥, 𝜔
(3.12)
Provided that each member exists.
In view of the equation (2.5) and (2.6) and the well known identity (cf.[10], p. 142, Eq. (18); see also
[8], p.172, prob.22(ii))
𝑒−𝑡
𝐿 𝑁
𝛼
(𝑥 + 𝑡) =
∞
𝑛=0
𝐿 𝑁
𝛼+𝑛
(𝑥)
(−𝑡) 𝑛
𝑛!
(3.13)](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-5-320.jpg)
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which follows immediately from the Taylor's expansion, since
𝐷𝑥
𝑛
{𝑒−𝑥
𝐿 𝑁
𝛼
(𝑥)} = (−1) 𝑛
𝑒−𝑥
𝐿 𝑁
𝛼+𝑛
(𝑥) (3.14)
Thus, above theorem 3.2 yields,
Corollary 3.2.1: If
Φ 𝑁 𝑥, 𝑦, 𝑡 =
∞
𝑛=0
𝑎 𝑛 𝑀𝑞𝑛
𝜇,𝜈,𝛾,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝐿 𝑁
𝜆+𝑛
𝑦 𝑡 𝑛
, (𝑎 𝑛 ≠ 0) (3.15)
and
Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥) =
𝑚𝑖𝑛 {𝑚,𝑛}
𝑘=0
(−1) 𝑘
𝑘!
𝑚 + 𝑛 − 2𝑘
𝑛 − 𝑘
𝑎 𝑚−𝑘 𝑍 𝑚−𝑘
𝜔 𝑛−𝑘
𝑀𝑞 𝑚+𝑛−2𝑘
𝜇,𝜈,𝛾,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 , (3.16)
then,
∞
𝑚,𝑛=0
Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 − 𝑘𝜔)−
𝜉+𝜆
𝑘 𝑒−𝑡
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝜔)−
1
𝑘 } 𝑟]
Φ 𝑁 𝑥 1 − 𝑘𝜔)−
1
𝑘 , 𝑦 + 𝑡,
𝑧𝑡
1 − 𝑘𝜔
, (𝑘 ≠ 0). (3.17)
Corollary 3.2.2: If
Ξ 𝑁 𝑥, 𝑦, 𝑡 =
∞
𝑛=0
𝑎 𝑛 𝑀𝑞𝑛
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝐿 𝑁
𝜆+𝑛
𝑦 𝑡 𝑛
, (𝑎 𝑛 ≠ 0) (3.18)
and
Λ 𝑚 ,𝑛 (𝑧, 𝜔, 𝑥) =
𝑚𝑖𝑛 {𝑚,𝑛}
𝑘=0
(−1) 𝑘
𝑘!
𝑚 + 𝑛 − 2𝑘
𝑛 − 𝑘
𝑎 𝑚−𝑘 𝑍 𝑚−𝑘
𝜔 𝑛−𝑘
𝑀𝑞 𝑚+𝑛−2𝑘
𝜇,𝜈,𝛾,𝜉−𝑘 𝑚+𝑛−2𝑘
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 , (3.19)
then,
∞
𝑚,𝑛=0
Λ 𝑚,𝑛 𝑧, 𝜔, 𝑥 𝐿 𝑁
𝜆+𝑚
𝑦 𝑡 𝑚
= (1 + 𝑘𝜔)−1+(𝜉+𝜆)/𝑘
𝑒−𝑡
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 + 𝑘𝜔)1/𝑘 } 𝑟]
Ξ 𝑁 𝑥(1 + 𝑘𝜔)
1
𝑘 , 𝑦 + 𝑡,
𝑧𝑡
1 + 𝑘𝜔
) , 𝑘 ≠ 0 . (3.20)
Particular cases of Corollary 3.2.1
(i) By using (1.7), which in conjunction with (3.6) yields the result obtained by Hubble and Srivastava [9],
∞
𝑚,𝑛=0
Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 − 𝑘𝜔)−𝛼/𝑘
exp 𝑝𝑥 𝑟
{1 − (1 − 𝑘𝜔)−𝑟/𝑘
} − 𝑡 Φ 𝑁 𝑥(1 − 𝑘𝜔)−1/𝑘
, 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑘𝜔) . (3.21)
(ii) By using (1.8), which in conjunction with the generating function (cf. [3]; p. 348, eq. (2.8)),
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝜏 𝑚+𝑛
𝛼
𝑥; 𝑟, 𝛽, 𝑘, 𝜂 𝑡 𝑛
= (1 − 𝑘𝑡)−𝑚−(𝛼+𝜂)/𝑘
exp 𝛽𝑥 𝑟
{1 − (1 − 𝑘𝑡)−𝑟/𝑘
} 𝜏 𝑚
(𝛼)
{𝑥(1 − 𝑘𝑡)−
1
𝑘 ; 𝑟, 𝛽, 𝑘, 𝜂}, (3.22)
we get
∞
𝑚,𝑛=0
Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 − 𝑘𝜔)−(𝛼+𝜂)/𝑘
exp 𝛽𝑥 𝑟
{1 − (1 − 𝑘𝜔)−𝑟/𝑘
} − 𝑡 Φ 𝑁 𝑥(1 − 𝑘𝜔)−1/𝑘
, 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑘𝜔) . (3.23)
(iii) By using (1.9), which in conjunction with (3.8), yields the generating relation
∞
𝑚,𝑛=0
Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 − 𝑎𝜔)−(𝛿+𝑠)/𝑎
𝑒−𝑡
𝐸 𝛼,𝛽
𝛾,𝑞
{𝑝 𝑘 (𝑥)}
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 {𝑥(1 − 𝑎𝜔)−1/𝑎 }
Φ 𝑁 𝑥(1 − 𝑎𝜔)−1/𝑎
, 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑎𝜔) (3.24)](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-6-320.jpg)
![A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel …
DOI: 10.9790/5728-11165563 www.iosrjournals.org 61 | Page
Particular cases of Corollary 3.2.2:
(i) By using (1.7), which in conjunction with the generating function (cf. [1]; p. 239)
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝐺 𝑚+𝑛
𝛼−𝑘𝑛
(𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛
= (1 + 𝑘𝑡)−1+
𝛼
𝑘 exp 𝑝𝑥 𝑟
{1 − (1 + 𝑘𝑡)
𝑟
𝑘 } 𝐺 𝑚
𝛼
{𝑥(1 + 𝑘𝑡)
1
𝑘 , 𝑟, 𝑝, 𝑘}, (3.25)
we get the result obtained by Hubble and Srivastava [9],
∞
𝑚,𝑛=0
Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 + 𝑘𝜔)−1+𝛼/𝑘
exp 𝑝𝑥 𝑟
{1 − (1 + 𝑘𝜔) 𝑟/𝑘
} − 𝑡 Ξ 𝑁 𝑥(1 + 𝑘𝜔)1/𝑘
, 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑘𝜔) (3.26)
(ii) By using (1.8), which in conjunction with the generating function ([3]; p. 359),
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝜏 𝑚+𝑛
(𝛼−𝑘𝑛)
(𝑥; 𝑟, 𝛽, 𝑘, 𝜂)𝑡 𝑛
= (1 + 𝑘𝑡)−1+(𝛼+𝜂)/𝑘
exp 𝛽𝑥 𝑟
{1 − (1 + 𝑘𝑡) 𝑟/𝑘
} 𝜏 𝑚
(𝛼)
{𝑥(1 + 𝑘𝑡)
1
𝑘 ; 𝑟, 𝛽, 𝑘, 𝜂}, (3.27)
we obtain,
∞
𝑚,𝑛=0
Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 + 𝑘𝜔)−1+(𝛼+𝜂)/𝑘
exp 𝛽𝑥 𝑟
{1 − (1 + 𝑘𝜔) 𝑟/𝑘
} − 𝑡 Ξ 𝑁 𝑥(1 + 𝑘𝜔)1/𝑘
, 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑘𝜔) . (3.28)
(iii) By using (1.9), which in conjunction with the generating function ([2]; p. 26),
∞
𝑛=0
𝑚 + 𝑛
𝑛
𝐴 𝑞(𝑚+𝑛)
(𝛼,𝛽,𝛾,𝛿−𝑎𝑛 )
(𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑛
= (1 + 𝑎𝑡)−1+
𝛿+𝑠
𝑎
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 𝑥
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 {𝑥(1 + 𝑎𝑡)
1
𝑎 }
𝐴 𝑞𝑚
𝛼,𝛽,𝛾,𝛿
𝑥(1 + 𝑎𝑡)
1
𝑎 ; 𝑎, 𝑘, 𝑠 , (3.29)
we obtain,
∞
𝑚,𝑛=0
Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁
(𝜆+𝑚)
(𝑦)𝑡 𝑚
= (1 + 𝑎𝜔)−1+(𝛿+𝑠)/𝑎
𝑒−𝑡
𝐸 𝛼,𝛽
𝛾,𝑞
{𝑝 𝑘 (𝑥)}
𝐸 𝛼,𝛽
𝛾,𝑞
𝑝 𝑘 {𝑥(1 + 𝑎𝜔)1/𝑎 }
Ξ 𝑁 𝑥(1 + 𝑎𝜔)1/𝑎
, 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑎𝜔) . (3.30)
The results (3.23), (3.24), (3.28), and (3.30) are believed to be new.
Further we use to recall here the theorem of (see [4]) to obtain another bilateral generating relation as
follows:
Theorem 3.3:If sequence {𝛥𝜇 (𝑥): 𝜇 is a complex number} is generated by
∞
𝑛=0
𝛾𝜇,𝑛 Δ𝜇+𝑛 (𝑥)𝑡 𝑛
= 𝜃(𝑥, 𝑡){𝜙(𝑥, 𝑡)}−𝜇
Δ𝜇 (𝜓(𝑥, 𝑡)). (3.31)
where 𝛾𝜇,𝑛 are arbitrary constants and 𝜃, 𝜙 and 𝜓 are arbitrary functions of 𝑥 and 𝑡.
Let
Φ 𝑞,𝑣 𝑥, 𝑡 =
∞
𝑛=0
𝛿 𝑣,𝑛 Δ 𝑣+𝑞𝑛 𝑥 𝑡 𝑛
, 𝛿 𝑣,𝑛 ≠ 0 (3.32)
𝑞 is a positive integer and 𝑣 is an arbitrary complex number, then
∞
𝑛=0
Δ𝜇+𝑛 (𝑥)𝑅 𝑛,𝑣
𝑞
(𝑦)𝑡 𝑛
= 𝜃(𝑥, 𝑡){𝜙 𝑥, 𝑡 }−𝜇
Φ 𝑞,𝑣 𝜓 𝑥, 𝑡 , 𝑦
𝑡
𝜙 𝑥, 𝑡
𝑞
(3.33)
where 𝑅 𝑛,𝑣
𝑞
(𝑦) is a polynomial of degree [𝑛/𝑞] in 𝑦, which is defined as
𝑅 𝑛,𝑣
𝑞
(𝑦) =
𝑛
𝑞
𝑘=0
𝛾 𝑣+𝑞𝑘,𝑛−𝑞𝑘 𝛿 𝑣,𝑘 𝑦 𝑘
(3.34)
In view of the relation (2.5), if 𝜇 = 𝑚, 𝛾 =
𝑚 + 𝑛
𝑛
, Δ 𝑚 (𝜓(𝑥, 𝑡) = 𝑀𝑞𝑚
𝜇,𝜈,𝛾,𝜉
{𝑥(1 − 𝑘𝑡)−
1
𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆},
𝜙(𝑥, 𝑡) = 1, 𝜃(𝑥, 𝑡) = (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)/𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑡)−1/𝑘
} 𝑟
]and 𝜓(𝑥, 𝑡) = 𝑥(1 − 𝑘𝑡)−1/𝑘
.](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-7-320.jpg)
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Then,
∞
𝑛=0
𝑀𝑞 𝑣+𝑛
𝜇,𝜈,𝛾,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑅 𝑛,𝑣
𝑏
𝑦 𝑡 𝑛
= (1 − 𝑘𝑡)−𝑚−
𝜉+𝜆
𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑡)−
1
𝑘 } 𝑟]
Φ 𝑏,𝑦 𝑥(1 − 𝑘𝑡)−
1
𝑘 , 𝑦𝑡 𝑏
(3.35)
where,
Φ 𝑏,𝑣 𝑥, 𝑡 =
𝑛=0
δ 𝑣,𝑛 𝑀𝑞 𝑣+𝑛
𝜇,𝜈,𝛾,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛
, 𝛿 𝑣,𝑛 ≠ 0
𝑅 𝑛,𝑣
𝑏
(𝑦) =
[𝑛/𝑏]
𝑘=0
𝑣 + 𝑛
𝑣 + 𝑏𝑘
𝛿 𝑣,𝑘 𝑦 𝑘
,
is a polynomial of degree [𝑛/𝑏] in 𝑦 and 𝑏 is a positive integer and 𝑣 is an arbitrary complex number.
IV. Generating Functions Involving Stirling Number of Second Kind
In his work Riodan [11], denoted a Stirling number of second kind by 𝑆(𝑛, 𝑘) and is defined as
𝑆(𝑛, 𝑘) =
1
𝑘!
𝑘
𝑗 =1
(−1) 𝑘−𝑗 𝑘
𝑗
𝑗 𝑛
(4.1)
so that,
𝑆 𝑛, 0 =
1, 𝑛 = 0
0, 𝑛 ∈ 𝑁
(4.2)
𝑆 𝑛, 1 = 𝑆 𝑛, 𝑛 = 1 𝑎𝑛𝑑 𝑆 𝑛, 𝑛 − 1 =
𝑛
2
. (4.3)
Recently, several authors have developed a number of families of generating function associated with Stirling
number of second kind 𝑆(𝑛, 𝑘) defined by (4.1).
To derive the generating function for 𝑀𝑞𝑛
(𝜇,𝜈,𝛾,ξ)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆) defined by (1.5), we use the theorem of
Srivastava [12] as
Theorem 4.1:Let the sequence {𝜉 𝑛 (𝑥)} 𝑛=0
∞
be generated by
∞
𝑘=0
𝑛 + 𝑘
𝑘
𝜉 𝑛+𝑘 (𝑥)𝑡 𝑘
= 𝑓(𝑥, 𝑡){𝑔 𝑥, 𝑡 }−𝑛
𝜉𝑛 𝑥, 𝑡 (4.4)
where 𝑓, 𝑔 and are suitable functions of 𝑥 and 𝑡.
Then, in terms of the Stirling number 𝑆(𝑛, 𝑘) defined by equation (4.1), the following family of generating
function
∞
𝑘=0
𝑘 𝑛
𝜉 𝑘((𝑥, −𝑧))
𝑧
𝑔 𝑥, −𝑧
𝑘
= {𝑓 𝑥, −𝑧 }−1
𝑛
𝑘=0
𝑘! 𝑆 𝑛, 𝑘 𝜉 𝑥 𝑧 𝑘
(4.5)
holds true provided that each member of equation (4.5) exists.
The generating function (2.5) and (2.6) relates to the family given by (4.4). Now by comparing (2.5) and
(4.4), it is easily observed that
𝑓(𝑥, 𝑡) = (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟
)
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑡)−1/𝑘 } 𝑟]
𝑔(𝑥, 𝑡) = (1 − 𝑘𝑡), (𝑥, 𝑡) = 𝑥(1 − 𝑘𝑡)−
1
𝑘
and
𝜉 𝑘 (𝑥) = 𝑀𝑞𝑘
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, λ)
Then the equation (4.5) of theorem 4.1, yields the generating function
∞
𝑘=0
𝑀𝑞𝑘
(𝜇,𝜈,𝛾,𝜉)
(𝑥(1 + 𝑘𝑧)−1/𝑘
; 𝑟, 𝑝, 𝑘, 𝜆)
𝑧
1 + 𝑘𝑧
𝑘
= (1 + 𝑘𝑧)(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 + 𝑘𝑧)−1/𝑘
} 𝑟
]
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟 )
𝑛
𝑘=0
𝑘! 𝑆 𝑛, 𝑘 𝑀𝑞𝑘
𝜇,𝜈,𝛾,𝜉
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 𝑘
(4.6)
(𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1
; 𝑘 ≠ 0)
Replacing 𝑧 by
𝑧
1−𝑘𝑧
and 𝑥 by
𝑥
(1−𝑘𝑧)1/𝑘the above equation immediately yields,](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-8-320.jpg)
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∞
𝑘=0
𝑀𝑞𝑘
(𝜇,𝜈,𝛾,𝜉)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑧 𝑘
= (1 − 𝑘𝑧)
−𝜉+𝜆
𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑧)−
1
𝑘 } 𝑟]
𝑛
𝑘=0
𝑘! 𝑆(𝑛, 𝑘)𝑀𝑞𝑘
𝜇,𝜈,𝛾,𝜉
x 1 − 𝑘𝑧 −
1
𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆
𝑧
1 − 𝑘𝑧
𝑘
(4.7)
(𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1
; 𝑘 ≠ 0)
Similarly, theorem 4.1, applied to the generating function (2.6), would readily gives us the generating function
∞
𝑘=0
𝑘 𝑛
𝑀𝑞𝑘
(𝜇,𝜈,𝛾,𝜉−𝑘𝑛)
(𝑥(1 − 𝑘𝑧)1/𝑘
; 𝑟, 𝑝, 𝑘, 𝜆)
𝑧
1 − 𝑘𝑧
𝑘
= (1 − 𝑘𝑧)1−(𝜉+𝜆)/𝑘
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 − 𝑘𝑧)1/𝑘
} 𝑟
]
𝐽𝜈,𝑞
𝜇,𝛾
(−𝑝𝑥 𝑟)
𝑛
𝑘=0
𝑘! 𝑆 𝑛, k 𝑀𝑞𝑘
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 𝑘
(4.8)
(𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1
; 𝑘 ≠ 0)
Thus for replacing𝑧 by
𝑧
1+𝑘𝑧
and 𝑥 by
𝑥
(1+𝑘𝑧)1/𝑘 the above equation immediately yields,
∞
𝑘=0
𝑘 𝑛
𝑀𝑞𝑘
(𝜇,𝜈,𝛾,𝜉−𝑘𝑛)
(𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑧 𝑘
= (1 + 𝑘𝑧)−1+
𝜉+𝜆
𝑘
𝐽𝜈,𝑞
𝜇,𝛾
−𝑝𝑥 𝑟
𝐽𝜈,𝑞
𝜇,𝛾
[−𝑝{𝑥(1 + 𝑘𝑧)
1
𝑘 } 𝑟]
𝑛
𝑘=0
𝑘! 𝑆 𝑛, 𝑘 𝑀𝑞𝑘
𝜇,𝜈,𝛾,𝜉−𝑘𝑛
x 1 + 𝑘𝑧
1
𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆
𝑧
1 + 𝑘𝑧
𝑘
(4.9)
(𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1
; 𝑘 ≠ 0)
References
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Simulation,2 (2001), 83-88.
[3] K.Y. Chen, C.J. Chyan, and H.M. Srivastava, Some polynomials associated with a certain family of differential operator, Journal of
Mathematical Analysis and Applications,268 (2002), 344-377.
[4] A.K. Shukla and J.C. Prajapati, A general class of polynomials associated with generalized Mittag- Leffler function , Integral
Transform and Special Function, Vol.19, No.1 (2008), 23-34.
[5] A.K. Shukla, and J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl.,336 (2007),
797-811.
[6] M. Singh, M.A. Khan, and A.H. Khan, On some properties of generalized Bessel-Maitland function, International Journal of
Mathematics Trends and Technology, Volume 14, Number 1, Oct 2014.
[7] K.R. Patil, and N.K. Thakre, Operational formulas for the function defined by a generalized Rodrigue's formula-II, Shivaji Univ.
J.15 (1975), 1-10.
[8] H.M. Srivastava, and H. L. Manocha, A Treatise on Generating Functions(Halsted Press (Ellis Horwood Limited, Chichester), John
Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984).
[9] J.H. Hubble, and H.M. Srivastava, Certain theorem on bilateral generating functions involving Hermite, Laguerre and Gegenbauer
polynomials, Journal of Mathematical Analysis and Applications, 152 (1990), 343-353.
[10] H. Buchholz, The confluent hypergeometric function (Translated from the German by H. Lichtblau and K. Wetzel, Springer tract in
Natural Philosophy, vol. 15, Springer-Verlag, New York, 1969).
[11] J. Riodan, Combinatorial Identities, (New York/London/Sydney; Wiley, 1968).
[12] H.M. Srivastava, Some families of generating functions associated with Stirling number of second kind, Journal of Mathematical
Analysis and Applications,251 (2000), 752-759.](https://image.slidesharecdn.com/h011165563-151126041702-lva1-app6892/85/A-Class-of-Polynomials-Associated-with-Differential-Operator-and-with-a-Generalization-of-Bessel-Maitland-Function-9-320.jpg)
A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel-Maitland Function
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. VI (Jan - Feb. 2015), PP 55-63 www.iosrjournals.org DOI: 10.9790/5728-11165563 www.iosrjournals.org 55 | Page A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel-Maitland Function Manoj Singh1 , Mumtaz Ahmad Khan2 and Abdul Hakim Khan3 1 (Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia) 2,3 (Department of Applied Mahematics, Faculty of Engineering, Aligarh Muslim University, U.P., India) Abstract:The object of this paper is to present several classes of linear and bilateral generating relations by employing operational techniques, which reduces as a special case of (known or new) bilateral generating relations.At last some of generating functions associated with stirling number of second kind are also discussed. Mathematics Subject Classification(2010): Primary 42C05, Secondary 33C45. Keywords: Generating relations, Differential operator, Rodrigue formula, Stirling number. I. Introduction In literature several authors have discussed a number of polynomials defined by their Rodrigue's formula and gave several classes of linear, bilinear, bilateral and mixed multilateral generating functions. In 1971, H.M. Srivastava and J.P.Singhal [1], introduced a general class of polynomial 𝐺𝑛 (𝛼) (𝑥, 𝑟, 𝑝, 𝑘) (of degree 𝑛 in 𝑥 𝑟 and in 𝛼) as 𝐺𝑛 𝛼 (𝑥, 𝑟, 𝑝, 𝑘) = 𝑥−𝛼−𝑘𝑛 𝑛! exp(𝑝𝑥 𝑟 )(𝑥 𝑘+1 𝐷) 𝑛 {𝑥 𝛼 exp(−𝑝𝑥 𝑟 )} (1.1) where Laguerre, Hermite, and Konhauser plynomials are the special case of (1.1). In 2001, H.M. Srivastava [2], defined a polynomial set of degree 𝑛 in 𝑥 𝑟 , 𝛼 and 𝜂 by 𝜏 𝑛 𝛼 (𝑥; 𝑟, 𝛽, 𝑘, 𝜂) = 𝑥−𝛼−𝑘𝑛 𝑛! exp(𝛽𝑥 𝑟 ){𝑥 𝑘 𝜂 + 𝑥𝐷 } 𝑛 𝑥 𝛼 exp −𝛽𝑥 𝑟 , 𝑛 ∈ 𝑁0 (1.2) A detailed account of (1.2) is present in the work of Chen et al. [3], who derived several new classes of linear, bilinear and mixed multilateral generating relations. Recently in 2008, Shukla and Prajapati [4], introduced a class of polynomial which is connected by the generalized Mittag-Leffler function 𝐸 𝛼,𝛽 𝛾,𝑞 (𝑧), whose properties were discussed in his paper [5], is defined as 𝐴 𝑞𝑛 𝛼,𝛽,𝛾,𝑞 (𝑥; 𝑎, 𝑘, 𝑠) = 𝑥−𝛿−𝑎𝑛 𝑛! 𝐸 𝛼,𝛽 𝛾,𝑞 {𝑝 𝑘 (𝑥)}{𝑥 𝑎 𝑠 + 𝑥𝐷 } 𝑛 𝑥 𝛿 𝐸 𝛼,𝛽 𝛾,𝑞 −𝑝 𝑘 𝑥 (1.3) where 𝛼, 𝛽, 𝛾, 𝛿 are real or complex numbers and 𝑎, 𝑘, 𝑠 are constants. In the present paper we have defined a polynomial by using the differential operator 𝜃, which involves two parameters 𝑘 and 𝜆 independent of 𝑥 as 𝜃 = 𝑥 𝑘 𝜆 + 𝑥𝐷𝑥 , 𝐷𝑥 = 𝑑 𝑑𝑥 (1.4) In connection with the differential operator (1.4), we define the polynomial 𝑀𝑞𝑛 (𝜇,𝑛𝑢,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) of degree 𝑛 in 𝑥 𝑟 , 𝜉 and 𝜆 as 𝑀𝑞𝑛 𝜇,𝜈,𝛾,𝜉 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝑥−𝜉−𝑘𝑛 𝑛! 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 )𝜃 𝑛 {𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 (𝑝𝑥 𝑟 )} (1.5) In (1.5), 𝜇, 𝜈, 𝛾, 𝜉 are real or complex numbers and 𝐽𝜈,𝑞 𝜇,𝛾 (𝑧) is the generalized Bessel's Maitland function for 𝜇, 𝜈, 𝛾 ∈ 𝐶; 𝑅𝑒(𝜇) ≥ 0, 𝑅𝑒(𝜈) ≥ 0, 𝑅𝑒(𝛾) ≥ 0 and 𝑞 ∈ (0,1) ∪ 𝑁 as 𝐽𝜈,𝑞 𝜇,𝛾 (𝑧) = ∞ 𝑛=0 (𝛾) 𝑞𝑛 (−𝑧) 𝑛 𝑛! Γ 𝜇𝑛 + 𝜈 + 1 (1.6) where (𝛾) 𝑞𝑛 = Γ(𝛾+𝑞𝑛) Γ(𝛾) denotes the generalized Pochhammer symbol, which reduces to 𝑞 𝑞𝑛 𝑞 𝑟=1 𝛾+𝑟−1 𝑞 𝑛 , if 𝑞 ∈ 𝑁. Certain properties of (1.6) have been discussed in [6]. Some special cases of the polynomial (1.5) are given below: 𝑀 𝑛 1,0,1,𝛼 (𝑥; 𝑟, 𝑝, 𝑘, 0) = 𝐺𝑛 𝛼 (𝑥, 𝑟, 𝑝, 𝑘) (1.7) where 𝐺𝑛 (𝛼) (𝑥, 𝑟, 𝑝, 𝑘) is defined by equation (1.1). 𝑀 𝑛 1,0,1,𝛼 (𝑥; 𝑟, 𝛽, 𝑘, 𝜂) = 𝜏 𝑛 𝛼 (𝑥; 𝑟, 𝛽, 𝑘, 𝜂) (1.8) where𝜏 𝑛 (𝛼) (𝑥; 𝑟, 𝛽, 𝑘, 𝜂) is defined by equation (1.2).
- 2. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 56 | Page 𝑀𝑞𝑛 𝛼,𝛽−1,𝛾,𝛿 (𝑥; 𝑘, 𝑝, 𝑎, 𝑠) = 𝐴 𝑞𝑛 𝛼,𝛽,𝛾,𝑞 (𝑥; 𝑎, 𝑘, 𝑠) (1.9) where 𝐴 𝑞𝑛 (𝛼,𝛽,𝛾,𝑞) (𝑥; 𝑎, 𝑘, 𝑠) is defined by equation (1.3). II. Linear Generating Functions By appealing the property of the operator of Patil and Thakre [7], 𝑒 𝑡𝜃 {𝑥 𝛼 𝑓(𝑥)} = 𝑥 𝛼 (1 − 𝑡𝑘𝑥 𝑘 ) 𝛼+𝜆 𝑘 . 𝑓 𝑥 (1 − 𝑡𝑘𝑥 𝑘 ) 1 𝑘 (2.1) and 𝑒 𝑡𝜃 {𝑥 𝛼−𝑛 𝑓(𝑥)} = 𝑥 𝛼 (1 + 𝑘𝑡)−1+ 𝛼+𝜆 𝑘 𝑓 𝑥(1 + 𝑘𝑡) 1 𝑘 (2.2) one obtains certain generating relations for the polynomials defined by (1.5) as given below: ∞ 𝑛=0 𝑀𝑞𝑛 𝜇,𝜈,𝛾,𝜉 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 − 𝑘𝑡)− 𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 𝑝 𝑥 1 − 𝑘𝑡 − 1 𝑘 𝑟 (2.3) (|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) ∞ 𝑛=0 𝑀𝑞𝑛 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 )𝐽𝜈,𝑞 𝜇,𝛾 [𝑝{𝑥 1 + 𝑘𝑡) 1 𝑘 } 𝑟 (2.4) (|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝑀𝑞(𝑚+𝑛) (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑡)− 1 𝑘 } 𝑟] 𝑀𝑞𝑚 𝜇,𝜈,𝛾,𝜉 (𝑥 1 − 𝑘𝑡)− 1 𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆 (2.5) (𝑛 ∈ 𝑁0;|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝑀𝑞(𝑚+𝑛) (𝜇,𝜈,𝛾,𝜉−𝑘𝑛) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 + 𝑘𝑡)−1+(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 + 𝑘𝑡)1/𝑘 } 𝑟] 𝑀𝑞𝑚 (𝜇,𝜈,𝛾,𝜉) (𝑥(1 + 𝑘𝑡)1/𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆) (2.6) (𝑛 ∈ 𝑁0;|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) Proof of (2.5): From (1.5), we write ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝑀𝑞 𝑚+𝑛 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛 = ∞ 𝑛=0 𝑥−𝜉− 𝑚+𝑛 𝑘 𝑚! 𝑛! 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝜃 𝑚+𝑛 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 𝑝𝑥 𝑟 𝑡 𝑛 = 𝑥−𝜉−𝑚𝑘 𝑚! 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 ∞ 𝑛=0 𝑥−𝑛𝑘 𝑡 𝑛 𝜃 𝑛 𝑛! . 𝜃 𝑚 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 𝑝𝑥 𝑟 = 𝑥−𝜉−𝑚𝑘 𝑚! 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 exp 𝑥−𝑘 𝑡𝜃 𝜃 𝑚 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 𝑝𝑥 𝑟 Again by using (1.5) and (2.1), we obtain the generating relation (2.5). Proof of (2.6): Multiplying equation (2.4) by 𝑥 𝜉 𝐽 𝜈,𝑞 𝜇 ,𝛾 (−𝑝𝑥 𝑟) and then operating upon both sides by the differential operator 𝜃 𝑚 , we get ∞ 𝑛=0 𝜃 𝑚 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉−𝑘𝑛) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) 𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝜉+𝜆 𝑘 𝜃 𝑚 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 [𝑝{𝑥(1 + 𝑘𝑡) 1 𝑘 } 𝑟 ] (2.7) Now replacing 𝑛 by 𝑚 in (1.5), we obtain 𝑚! 𝑥 𝜉+𝑚𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝑀𝑞𝑚 𝜇,𝜈,𝛾,𝜉 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝜃 𝑚 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 𝑝𝑥 𝑟 (2.8) Again replacing 𝑚 by 𝑚 + 𝑛 in equation (2.8), we obtain (𝑚 + 𝑛)! 𝑥 𝜉+𝑚𝑘 +𝑛𝑘 𝐽 𝜈,𝑞 𝜇 ,𝛾 −𝑝𝑥 𝑟 𝑀𝑞 𝑚+𝑛 𝜇,𝜈,𝛾,𝜉 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝜃 𝑚+𝑛 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 𝑝𝑥 𝑟
- 3. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 57 | Page = 𝜃 𝑚 𝜃 𝑛 {𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 (𝑝𝑥 𝑟 )} = 𝑛! 𝑥 𝜉+𝑛𝑘 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) (2.9) Further replacing 𝜉 by (𝜉 − 𝑘𝑛) in (2.9), we obtain 𝑥 𝜉 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟) 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉−𝑘𝑛) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) = 𝑚 + 𝑛! 𝑛! 𝑥 𝜉+𝑚𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝑀𝑞 𝑚+𝑛 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 (2.10) Substituting the value of equation (2.10) in equation (2.7) and using equation (2.8), we obtain ∞ 𝑛=0 𝑚 + 𝑛! 𝑛! 𝑥 𝜉+𝑘𝑛 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝑀𝑞 𝑚+𝑛 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝜉+𝜆 𝑘 𝑚! 𝑥 𝜉+𝑘𝑛 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 + 𝑘𝑡) 1 𝑘 } 𝑟] 𝑀𝑞𝑚 𝜇,𝜈,𝛾,𝜉 x 1 + 𝑘𝑧 1 𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆 which gives the generating relation (2.6). The theorem by H.M. Srivastava on mixed generating functions (cf. [8], p. 378, Th. 12), plays an important role to derive several other generating relation for the polynomial 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) defined by (1.5). Theorem 2.1:Let each of the functions 𝐴(𝑧),𝐵𝑗 (𝑧)(𝑗 = 1, . . . , 𝑚), and 𝑧−1 𝐶𝑙(𝑧)(𝑙 = 1, . . . , 𝑠) be analytic in a neighborhood of the origin, and assume that 𝐴 0 . 𝐵 0 . 𝐶′ 𝑙 0 ≠ 0 𝑗 = 1, . . . , 𝑚; 𝑙 = 1, . . . , 𝑠 . (2.11) Define the sequence of function 𝑔 𝑛 (𝛼1,...,𝛼 𝑚 ) (𝑥1, . . . , 𝑥 𝑠) 𝑛=0 ∞ by 𝐴(𝑧) 𝑚 𝑗 =1 {[𝐵𝑗 (𝑧)] 𝛼 𝑗 }. exp 𝑠 𝑙=1 𝑥𝑙 𝐶𝑙(𝑧) = ∞ 𝑛=0 𝑔 𝑛 𝛼1,…,𝛼 𝑚 𝑥1, . . . , 𝑥 𝑠 𝑧 𝑛 𝑛! (2.12) where 𝛼1, . . . , , 𝛼 𝑚 and 𝑥1, . . . , 𝑥 𝑠 are arbitrary complex numbers independent of 𝑧. Then, for arbitrary parameters 𝜆1, . . . , 𝜆 𝑚 and 𝑦1,. . . , 𝑦𝑠 independent of z, ∞ 𝑛=0 𝑔 𝑛 (𝛼1+𝜆1 𝑛,...,𝛼 𝑚 +𝜆 𝑚 𝑛) (𝑥1 + 𝑛𝑦1, . . . , 𝑥 𝑠 + 𝑛𝑦𝑠) 𝑡 𝑛 𝑛! = 𝐴(𝜁) 𝑚 𝑗=1 {[𝐵𝑗 (𝜁)] 𝛼 𝑗 }. exp 𝑠 𝑙=1 𝑥𝑙 𝐶𝑙(𝜁) 1 − 𝜁 𝑚 𝑗=1 𝜆𝑗 [𝐵′𝑗 (𝜁)/𝐵𝑗 (𝜁)] + 𝑠 𝑙=1 𝑦𝑙 𝐶′𝑙(𝜁) 𝜁 = 𝑡 𝑚 𝑗 =1 {[𝐵𝑗 (𝜁)] 𝜆 𝑗 }exp 𝑠 𝑙=1 𝑦𝑙 𝐶𝑙(𝜁) ; 𝑚, 𝑠 ∈ 𝑁 (2.13) By putting 𝜇 = 1, 𝜈 = 0, 𝛾 = 1 and 𝑞 = 1 in the generating relation (2.3), reduces into the form ∞ 𝑛=0 𝑀 𝑛 1,0,1,𝜉 (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 − 𝑘𝑡)− 𝜉+𝜆 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝑘𝑡)− 𝑟 𝑘 } (2.14) The generating function (2.14) is equivalently of the type (2.12), with of course, 𝑚 − 1 = 𝑠 = 1, 𝐴(𝑧) = 1, 𝐵1(𝑧) = 𝐵2(𝑧) = (1 − 𝑘𝑡)−1/𝑘 , 𝑥1 = 𝑝, 𝐶1(𝑧) = 𝑝𝑥 𝑟 [1 − (1 − 𝑘𝑡)− 𝑟 𝑘 and 𝑔 𝑛 𝜉,𝜆 𝑝 = 𝑛! 𝑀 𝑛 1,0,1,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 ;(𝑛 ∈ 𝑁). Thus applying the above theorem to the generating function (2.14), one obtains ∞ 𝑛=0 𝑀 𝑛 (1,0,1,𝜉+𝛿𝑛) (𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛 = (1 − 𝑘𝜁)−(𝜉+𝜆)/𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝑘𝜁)−𝑟/𝑘 } 1 − 𝜁(1 − 𝑘𝜁)−1 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟(1 − 𝑘𝜁)−𝑟/𝑘 (2.15) where, 𝜁 = 𝑡(1 − 𝑘𝜁)−(𝛿+𝑣)/𝑘 exp 𝑢𝑥 𝑟 {1 − (1 − 𝑘𝜁)−𝑟/𝑘 } ; 𝑘 ≠ 0 By setting 𝜁 by 𝜁 𝑘 (𝑘 ≠ 0), in the generating relation (2.15), one obtains ∞ 𝑛=0 𝑀 𝑛 (1,0,1,𝜉+𝛿𝑛) (𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛 = (1 − 𝜁)−(𝜉+𝜆)/𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝜁)−𝑟/𝑘 } 1 − 𝑘−1 𝜁(1 − 𝜁)−1 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟 (1 − 𝜁)−𝑟/𝑘 (2.16) where, 𝜁 = 𝑘𝑡(1 − 𝜁)−(𝛿+𝑣)/𝑘 exp 𝑢𝑥 𝑟 {1 − (1 − 𝜁)−𝑟/𝑘 } ; 𝑘 ≠ 0 By replacing 𝜁 by 𝜁 1+𝜁 in the generating function (2.16), one immediately obtains the generating function
- 4. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 58 | Page ∞ 𝑛=0 𝑀 𝑛 (1,0,1,𝜉+𝛿𝑛) 𝑥; 𝑟, 𝑝 + 𝑢𝑛, 𝑘, 𝜆 + 𝑣𝑛 𝑡 𝑛 = (1 + 𝜁)(𝜉+𝜆)/𝑘 exp 𝑝𝑥 𝑟 {1 − (1 + 𝜁) 𝑟/𝑘 } 1 − 𝑘−1 𝜁 𝛿 + 𝑣 − 𝑢𝑟𝑥 𝑟(1 + 𝜁) 𝑟/𝑘 (2.17) where, 𝜁 = 𝑘𝑡(1 + 𝜁)1+(𝛿+𝑣)/𝑘 exp 𝑢𝑥 𝑟 {1 − (1 + 𝜁) 𝑟/𝑘 } ; 𝑘 ≠ 0 Again by setting 𝛿 = −𝑣 − 𝑘 and 𝑢 = 0 in (2.17), one obtains ∞ 𝑛=0 𝑀 𝑛 (1,0,1,𝜉−(𝑣+𝑘)𝑛) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆 + 𝑣𝑛)𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝜉+𝜆 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 + 𝑘𝑡) 𝑟 𝑘 } (2.18) (|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) when, 𝑣 = 0, (2.18) readily gives us ∞ 𝑛=0 𝑀 𝑛 (1,0,1,𝜉−𝑘𝑛 ) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝜉+𝜆 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 + 𝑘𝑡) 𝑟 𝑘 } (2.19) (|𝑡| < |𝑘|−1 ; 𝑘 ≠ 0) The generating relation obtained (2.19) is the particular case of (2.4) at 𝜇 = 1, 𝜈 = 0, 𝛾 = 1 and 𝑞 = 1. With the help of (1.8), the generating relation (2.15), (2.16), (2.17), (2.18), and (2.19) reduce to the generating function (2.16), (2.19), (2.22), (2.24) and (2.51) of Chen et al. [3]. III. Bilateral Generating Functions A class of function {𝑆𝑛 (𝑥), 𝑛 = 0,1,2, . . . } (see [8], p. 411) generated by ∞ 𝑛=0 𝐴 𝑚,𝑛 𝑆 𝑚+𝑛 𝑡 𝑛 = 𝑓(𝑥, 𝑡){𝑔(𝑥, 𝑡)}−𝑚 𝑆 𝑚 ((𝑥, 𝑡)) (3.1) where 𝑚 ≥ 0 is an integer, the coefficient 𝐴 𝑚,𝑛 are arbitrary constant and 𝑓, 𝑔, are suitable functions of 𝑥 and 𝑡. Numerous classes of bilinear (or bilateral) generating functions were obtained for the function 𝑆𝑛 (𝑥) generated by (3.1). Theorem 3.1: (cf. [8], p. 412, Th. 13) For the sequence {𝑆𝑛 (𝑥)} generated by (3.1), let 𝐹 𝑥, 𝑡 = ∞ 𝑛=0 𝑎 𝑛 𝑆𝑛 𝑥 𝑡 𝑛 (3.2) where 𝑎 𝑛 ≠ 0 are arbitrary constant, Then, 𝑓 𝑥, 𝑡 𝐹 𝑥, 𝑡 , 𝑦𝑡 𝑔 𝑥, 𝑡 = ∞ 𝑛=0 𝑆𝑛 𝑥 𝜎𝑛 𝑦 𝑡 𝑛 (3.3) where 𝜎𝑛 (𝑦) is a polynomial (of degree 𝑛 in 𝑦) defined by 𝜎𝑛 𝑦 = 𝑛 𝑘=0 𝑎 𝑘 𝐴 𝑘,𝑛−𝑘 𝑦 𝑘 (3.4) By using the theorem 3.1, we obtain the bilateral generating function for the polynomial (1.5) as ∞ 𝑛=0 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝜎𝑛 (𝑦)𝑡 𝑛 = (1 − 𝑘𝑡)− 𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑡)− 1 𝑘 } 𝑟] 𝐹 𝑥(1 − 𝑘𝑡)− 1 𝑘 , 𝑦𝑡(1 − 𝑘𝑡)−1 (3.5) where, 𝐹(𝑥, 𝑡) = ∞ 𝑛=0 (𝑎) 𝑛 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 Proof of (3.5):With suitable replacement in theorem 3.1, we obtain ∞ 𝑛=0 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝜎𝑛 (𝑦)𝑡 𝑛 = ∞ 𝑛=0 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) 𝑛 𝑚=0 𝑛 𝑚 (𝑎) 𝑚 (𝑦) 𝑚 𝑡 𝑛
- 5. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 59 | Page = ∞ 𝑚=0 (𝑎) 𝑚 (𝑦𝑡) 𝑚 ∞ 𝑛=0 𝑚 + 𝑛 𝑚 𝑀𝑞(𝑚+𝑛) (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑡 𝑛 Making use of generating relation (2.5), yields the bilateral generating function (3.5). Particular cases of 3.5: (i) If we put 𝜇 = 1, 𝜈 = 0, 𝛾 = 1, 𝜆 = 0, 𝑞 = 1, 𝜉 = 𝛼 and 𝑎 = 𝜇 in (3.5), which in conjunction with the generating function (cf. [1]), ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝐺 𝑚+𝑛 (𝛼) (𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛 = (1 − 𝑘𝑡)−𝑚− 𝛼 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝑘𝑡)− 𝑟 𝑘 } 𝐺 𝑚 𝛼 𝑥(1 − 𝑘𝑡)− 1 𝑘 , 𝑟, 𝑝, 𝑘 (3.6) and the use of (1.7), we get bilinear generating function obtained by Srivastava and Singhal [1], ∞ 𝑛=0 𝑛! 𝐺𝑛 𝛼 (𝑥, 𝑟, 𝑝, 𝑘)𝜎𝑛 (𝑦)𝑡 𝑛 = (1 − 𝑘𝑡)− 𝛼 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝑘𝑡)− 𝑟 𝑘 } 𝐹 𝑥(1 − 𝑘𝑡)− 1 𝑘 , 𝑦𝑡(1 − 𝑘𝑡)−1 (3.7) where, 𝐹(𝑥, 𝑡) = ∞ 𝑛=0 (𝜇) 𝑛 𝐺𝑛 (𝛼) (𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛 (ii) Again, putting 𝜇 = 𝛼, 𝜈 = 𝛽 − 1, 𝑝𝑥 𝑟 = 𝑝 𝑘 (𝑥), 𝜉 = 𝛿, 𝑘 = 𝑎, 𝜆 = 𝑠 and 𝑎 = 𝜇 in (3.5), which in conjunction with the generating function (cf. [4]; p. 26. Eq. (1.6)) ∞ 𝑚=0 𝑚 + 𝑛 𝑛 𝐴 𝑞(𝑚+𝑛) (𝛼,𝛽,𝛾,𝛿) (𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑚 = (1 − 𝑎𝑡)−𝑛− 𝛿+𝑠 𝑎 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 𝑥 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 {𝑥(1 − 𝑎𝑡)− 1 𝑎 } 𝐴 𝑞𝑛 𝛼,𝛽,𝛾,𝛿 𝑥(1 − 𝑎𝑡)− 1 𝑎 ; 𝑎, 𝑘, 𝑠 (3.8) and use of (1.9), we obtain a new bilateral generating function for the polynomial defined by Shukla and Prajapati [4] as ∞ 𝑛=0 𝐴 𝑞𝑛 (𝛼,𝛽,𝛾,𝛿) (𝑥; 𝑎, 𝑘, 𝑠)𝜎𝑛 (𝑦)𝑡 𝑛 = (1 − 𝑎𝑡)− 𝛿+𝑠 𝑎 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 𝑥 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 {𝑥(1 − 𝑎𝑡)− 1 𝑎 } 𝐹 𝑥(1 − 𝑎𝑡)− 1 𝑎 , 𝑦𝑡(1 − 𝑎𝑡)−1 (3.9) where, 𝐹(𝑥, 𝑡) = ∞ 𝑛=0 (𝜇) 𝑛 𝐴 𝑞𝑛 (𝛼,𝛽,𝛾,𝛿) (𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑛 In 1990, Hubble and Srivastava [9] generalize the theorem. Theorem 3.2:Corresponding to the function 𝑆𝑛 (𝑥), generated by (3.1), let Θ 𝑁 𝑥, 𝑦, 𝑡 = ∞ 𝑛=0 𝑎 𝑛 𝑆𝑛 𝑥 𝐿 𝑁 𝜆+𝑛 𝑦 𝑡 𝑛 , 𝑎 𝑛 ≠ 0 (3.10) where 𝜆 is an arbitrary (real or complex) parameter. Suppose also that 𝜃 𝑚,𝑛 𝑧, 𝜔, 𝑥 = 𝑚𝑖𝑛 {𝑚,𝑛} 𝑘=0 (−1) 𝑘 𝑘 𝑎 𝑚−𝑘 𝐴 𝑚−𝑘,𝑛−𝑘 𝑧 𝑚−𝑘 𝜔 𝑛−𝑘 𝑆 𝑚+𝑛−2𝑘 𝑥 (3.11) Then, ∞ 𝑚,𝑛=0 𝜃 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 𝜆+𝑚 (𝑦)𝑡 𝑚 = exp −𝑡 𝑓 𝑥, 𝜔 Θ 𝑁 𝑥, 𝜔 , 𝑦 + 𝑡, 𝑧𝑡 𝑔 𝑥, 𝜔 (3.12) Provided that each member exists. In view of the equation (2.5) and (2.6) and the well known identity (cf.[10], p. 142, Eq. (18); see also [8], p.172, prob.22(ii)) 𝑒−𝑡 𝐿 𝑁 𝛼 (𝑥 + 𝑡) = ∞ 𝑛=0 𝐿 𝑁 𝛼+𝑛 (𝑥) (−𝑡) 𝑛 𝑛! (3.13)
- 6. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 60 | Page which follows immediately from the Taylor's expansion, since 𝐷𝑥 𝑛 {𝑒−𝑥 𝐿 𝑁 𝛼 (𝑥)} = (−1) 𝑛 𝑒−𝑥 𝐿 𝑁 𝛼+𝑛 (𝑥) (3.14) Thus, above theorem 3.2 yields, Corollary 3.2.1: If Φ 𝑁 𝑥, 𝑦, 𝑡 = ∞ 𝑛=0 𝑎 𝑛 𝑀𝑞𝑛 𝜇,𝜈,𝛾,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝐿 𝑁 𝜆+𝑛 𝑦 𝑡 𝑛 , (𝑎 𝑛 ≠ 0) (3.15) and Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥) = 𝑚𝑖𝑛 {𝑚,𝑛} 𝑘=0 (−1) 𝑘 𝑘! 𝑚 + 𝑛 − 2𝑘 𝑛 − 𝑘 𝑎 𝑚−𝑘 𝑍 𝑚−𝑘 𝜔 𝑛−𝑘 𝑀𝑞 𝑚+𝑛−2𝑘 𝜇,𝜈,𝛾,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 , (3.16) then, ∞ 𝑚,𝑛=0 Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 − 𝑘𝜔)− 𝜉+𝜆 𝑘 𝑒−𝑡 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝜔)− 1 𝑘 } 𝑟] Φ 𝑁 𝑥 1 − 𝑘𝜔)− 1 𝑘 , 𝑦 + 𝑡, 𝑧𝑡 1 − 𝑘𝜔 , (𝑘 ≠ 0). (3.17) Corollary 3.2.2: If Ξ 𝑁 𝑥, 𝑦, 𝑡 = ∞ 𝑛=0 𝑎 𝑛 𝑀𝑞𝑛 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝐿 𝑁 𝜆+𝑛 𝑦 𝑡 𝑛 , (𝑎 𝑛 ≠ 0) (3.18) and Λ 𝑚 ,𝑛 (𝑧, 𝜔, 𝑥) = 𝑚𝑖𝑛 {𝑚,𝑛} 𝑘=0 (−1) 𝑘 𝑘! 𝑚 + 𝑛 − 2𝑘 𝑛 − 𝑘 𝑎 𝑚−𝑘 𝑍 𝑚−𝑘 𝜔 𝑛−𝑘 𝑀𝑞 𝑚+𝑛−2𝑘 𝜇,𝜈,𝛾,𝜉−𝑘 𝑚+𝑛−2𝑘 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 , (3.19) then, ∞ 𝑚,𝑛=0 Λ 𝑚,𝑛 𝑧, 𝜔, 𝑥 𝐿 𝑁 𝜆+𝑚 𝑦 𝑡 𝑚 = (1 + 𝑘𝜔)−1+(𝜉+𝜆)/𝑘 𝑒−𝑡 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 + 𝑘𝜔)1/𝑘 } 𝑟] Ξ 𝑁 𝑥(1 + 𝑘𝜔) 1 𝑘 , 𝑦 + 𝑡, 𝑧𝑡 1 + 𝑘𝜔 ) , 𝑘 ≠ 0 . (3.20) Particular cases of Corollary 3.2.1 (i) By using (1.7), which in conjunction with (3.6) yields the result obtained by Hubble and Srivastava [9], ∞ 𝑚,𝑛=0 Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 − 𝑘𝜔)−𝛼/𝑘 exp 𝑝𝑥 𝑟 {1 − (1 − 𝑘𝜔)−𝑟/𝑘 } − 𝑡 Φ 𝑁 𝑥(1 − 𝑘𝜔)−1/𝑘 , 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑘𝜔) . (3.21) (ii) By using (1.8), which in conjunction with the generating function (cf. [3]; p. 348, eq. (2.8)), ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝜏 𝑚+𝑛 𝛼 𝑥; 𝑟, 𝛽, 𝑘, 𝜂 𝑡 𝑛 = (1 − 𝑘𝑡)−𝑚−(𝛼+𝜂)/𝑘 exp 𝛽𝑥 𝑟 {1 − (1 − 𝑘𝑡)−𝑟/𝑘 } 𝜏 𝑚 (𝛼) {𝑥(1 − 𝑘𝑡)− 1 𝑘 ; 𝑟, 𝛽, 𝑘, 𝜂}, (3.22) we get ∞ 𝑚,𝑛=0 Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 − 𝑘𝜔)−(𝛼+𝜂)/𝑘 exp 𝛽𝑥 𝑟 {1 − (1 − 𝑘𝜔)−𝑟/𝑘 } − 𝑡 Φ 𝑁 𝑥(1 − 𝑘𝜔)−1/𝑘 , 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑘𝜔) . (3.23) (iii) By using (1.9), which in conjunction with (3.8), yields the generating relation ∞ 𝑚,𝑛=0 Ψ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 − 𝑎𝜔)−(𝛿+𝑠)/𝑎 𝑒−𝑡 𝐸 𝛼,𝛽 𝛾,𝑞 {𝑝 𝑘 (𝑥)} 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 {𝑥(1 − 𝑎𝜔)−1/𝑎 } Φ 𝑁 𝑥(1 − 𝑎𝜔)−1/𝑎 , 𝑦 + 𝑡, 𝑧𝑡/(1 − 𝑎𝜔) (3.24)
- 7. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 61 | Page Particular cases of Corollary 3.2.2: (i) By using (1.7), which in conjunction with the generating function (cf. [1]; p. 239) ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝐺 𝑚+𝑛 𝛼−𝑘𝑛 (𝑥, 𝑟, 𝑝, 𝑘)𝑡 𝑛 = (1 + 𝑘𝑡)−1+ 𝛼 𝑘 exp 𝑝𝑥 𝑟 {1 − (1 + 𝑘𝑡) 𝑟 𝑘 } 𝐺 𝑚 𝛼 {𝑥(1 + 𝑘𝑡) 1 𝑘 , 𝑟, 𝑝, 𝑘}, (3.25) we get the result obtained by Hubble and Srivastava [9], ∞ 𝑚,𝑛=0 Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 + 𝑘𝜔)−1+𝛼/𝑘 exp 𝑝𝑥 𝑟 {1 − (1 + 𝑘𝜔) 𝑟/𝑘 } − 𝑡 Ξ 𝑁 𝑥(1 + 𝑘𝜔)1/𝑘 , 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑘𝜔) (3.26) (ii) By using (1.8), which in conjunction with the generating function ([3]; p. 359), ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝜏 𝑚+𝑛 (𝛼−𝑘𝑛) (𝑥; 𝑟, 𝛽, 𝑘, 𝜂)𝑡 𝑛 = (1 + 𝑘𝑡)−1+(𝛼+𝜂)/𝑘 exp 𝛽𝑥 𝑟 {1 − (1 + 𝑘𝑡) 𝑟/𝑘 } 𝜏 𝑚 (𝛼) {𝑥(1 + 𝑘𝑡) 1 𝑘 ; 𝑟, 𝛽, 𝑘, 𝜂}, (3.27) we obtain, ∞ 𝑚,𝑛=0 Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 + 𝑘𝜔)−1+(𝛼+𝜂)/𝑘 exp 𝛽𝑥 𝑟 {1 − (1 + 𝑘𝜔) 𝑟/𝑘 } − 𝑡 Ξ 𝑁 𝑥(1 + 𝑘𝜔)1/𝑘 , 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑘𝜔) . (3.28) (iii) By using (1.9), which in conjunction with the generating function ([2]; p. 26), ∞ 𝑛=0 𝑚 + 𝑛 𝑛 𝐴 𝑞(𝑚+𝑛) (𝛼,𝛽,𝛾,𝛿−𝑎𝑛 ) (𝑥; 𝑎, 𝑘, 𝑠)𝑡 𝑛 = (1 + 𝑎𝑡)−1+ 𝛿+𝑠 𝑎 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 𝑥 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 {𝑥(1 + 𝑎𝑡) 1 𝑎 } 𝐴 𝑞𝑚 𝛼,𝛽,𝛾,𝛿 𝑥(1 + 𝑎𝑡) 1 𝑎 ; 𝑎, 𝑘, 𝑠 , (3.29) we obtain, ∞ 𝑚,𝑛=0 Λ 𝑚,𝑛 (𝑧, 𝜔, 𝑥)𝐿 𝑁 (𝜆+𝑚) (𝑦)𝑡 𝑚 = (1 + 𝑎𝜔)−1+(𝛿+𝑠)/𝑎 𝑒−𝑡 𝐸 𝛼,𝛽 𝛾,𝑞 {𝑝 𝑘 (𝑥)} 𝐸 𝛼,𝛽 𝛾,𝑞 𝑝 𝑘 {𝑥(1 + 𝑎𝜔)1/𝑎 } Ξ 𝑁 𝑥(1 + 𝑎𝜔)1/𝑎 , 𝑦 + 𝑡, 𝑧𝑡/(1 + 𝑎𝜔) . (3.30) The results (3.23), (3.24), (3.28), and (3.30) are believed to be new. Further we use to recall here the theorem of (see [4]) to obtain another bilateral generating relation as follows: Theorem 3.3:If sequence {𝛥𝜇 (𝑥): 𝜇 is a complex number} is generated by ∞ 𝑛=0 𝛾𝜇,𝑛 Δ𝜇+𝑛 (𝑥)𝑡 𝑛 = 𝜃(𝑥, 𝑡){𝜙(𝑥, 𝑡)}−𝜇 Δ𝜇 (𝜓(𝑥, 𝑡)). (3.31) where 𝛾𝜇,𝑛 are arbitrary constants and 𝜃, 𝜙 and 𝜓 are arbitrary functions of 𝑥 and 𝑡. Let Φ 𝑞,𝑣 𝑥, 𝑡 = ∞ 𝑛=0 𝛿 𝑣,𝑛 Δ 𝑣+𝑞𝑛 𝑥 𝑡 𝑛 , 𝛿 𝑣,𝑛 ≠ 0 (3.32) 𝑞 is a positive integer and 𝑣 is an arbitrary complex number, then ∞ 𝑛=0 Δ𝜇+𝑛 (𝑥)𝑅 𝑛,𝑣 𝑞 (𝑦)𝑡 𝑛 = 𝜃(𝑥, 𝑡){𝜙 𝑥, 𝑡 }−𝜇 Φ 𝑞,𝑣 𝜓 𝑥, 𝑡 , 𝑦 𝑡 𝜙 𝑥, 𝑡 𝑞 (3.33) where 𝑅 𝑛,𝑣 𝑞 (𝑦) is a polynomial of degree [𝑛/𝑞] in 𝑦, which is defined as 𝑅 𝑛,𝑣 𝑞 (𝑦) = 𝑛 𝑞 𝑘=0 𝛾 𝑣+𝑞𝑘,𝑛−𝑞𝑘 𝛿 𝑣,𝑘 𝑦 𝑘 (3.34) In view of the relation (2.5), if 𝜇 = 𝑚, 𝛾 = 𝑚 + 𝑛 𝑛 , Δ 𝑚 (𝜓(𝑥, 𝑡) = 𝑀𝑞𝑚 𝜇,𝜈,𝛾,𝜉 {𝑥(1 − 𝑘𝑡)− 1 𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆}, 𝜙(𝑥, 𝑡) = 1, 𝜃(𝑥, 𝑡) = (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 )/𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑡)−1/𝑘 } 𝑟 ]and 𝜓(𝑥, 𝑡) = 𝑥(1 − 𝑘𝑡)−1/𝑘 .
- 8. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 62 | Page Then, ∞ 𝑛=0 𝑀𝑞 𝑣+𝑛 𝜇,𝜈,𝛾,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑅 𝑛,𝑣 𝑏 𝑦 𝑡 𝑛 = (1 − 𝑘𝑡)−𝑚− 𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑡)− 1 𝑘 } 𝑟] Φ 𝑏,𝑦 𝑥(1 − 𝑘𝑡)− 1 𝑘 , 𝑦𝑡 𝑏 (3.35) where, Φ 𝑏,𝑣 𝑥, 𝑡 = 𝑛=0 δ 𝑣,𝑛 𝑀𝑞 𝑣+𝑛 𝜇,𝜈,𝛾,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑡 𝑛 , 𝛿 𝑣,𝑛 ≠ 0 𝑅 𝑛,𝑣 𝑏 (𝑦) = [𝑛/𝑏] 𝑘=0 𝑣 + 𝑛 𝑣 + 𝑏𝑘 𝛿 𝑣,𝑘 𝑦 𝑘 , is a polynomial of degree [𝑛/𝑏] in 𝑦 and 𝑏 is a positive integer and 𝑣 is an arbitrary complex number. IV. Generating Functions Involving Stirling Number of Second Kind In his work Riodan [11], denoted a Stirling number of second kind by 𝑆(𝑛, 𝑘) and is defined as 𝑆(𝑛, 𝑘) = 1 𝑘! 𝑘 𝑗 =1 (−1) 𝑘−𝑗 𝑘 𝑗 𝑗 𝑛 (4.1) so that, 𝑆 𝑛, 0 = 1, 𝑛 = 0 0, 𝑛 ∈ 𝑁 (4.2) 𝑆 𝑛, 1 = 𝑆 𝑛, 𝑛 = 1 𝑎𝑛𝑑 𝑆 𝑛, 𝑛 − 1 = 𝑛 2 . (4.3) Recently, several authors have developed a number of families of generating function associated with Stirling number of second kind 𝑆(𝑛, 𝑘) defined by (4.1). To derive the generating function for 𝑀𝑞𝑛 (𝜇,𝜈,𝛾,ξ) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆) defined by (1.5), we use the theorem of Srivastava [12] as Theorem 4.1:Let the sequence {𝜉 𝑛 (𝑥)} 𝑛=0 ∞ be generated by ∞ 𝑘=0 𝑛 + 𝑘 𝑘 𝜉 𝑛+𝑘 (𝑥)𝑡 𝑘 = 𝑓(𝑥, 𝑡){𝑔 𝑥, 𝑡 }−𝑛 𝜉𝑛 𝑥, 𝑡 (4.4) where 𝑓, 𝑔 and are suitable functions of 𝑥 and 𝑡. Then, in terms of the Stirling number 𝑆(𝑛, 𝑘) defined by equation (4.1), the following family of generating function ∞ 𝑘=0 𝑘 𝑛 𝜉 𝑘((𝑥, −𝑧)) 𝑧 𝑔 𝑥, −𝑧 𝑘 = {𝑓 𝑥, −𝑧 }−1 𝑛 𝑘=0 𝑘! 𝑆 𝑛, 𝑘 𝜉 𝑥 𝑧 𝑘 (4.5) holds true provided that each member of equation (4.5) exists. The generating function (2.5) and (2.6) relates to the family given by (4.4). Now by comparing (2.5) and (4.4), it is easily observed that 𝑓(𝑥, 𝑡) = (1 − 𝑘𝑡)−𝑚−(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑡)−1/𝑘 } 𝑟] 𝑔(𝑥, 𝑡) = (1 − 𝑘𝑡), (𝑥, 𝑡) = 𝑥(1 − 𝑘𝑡)− 1 𝑘 and 𝜉 𝑘 (𝑥) = 𝑀𝑞𝑘 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, λ) Then the equation (4.5) of theorem 4.1, yields the generating function ∞ 𝑘=0 𝑀𝑞𝑘 (𝜇,𝜈,𝛾,𝜉) (𝑥(1 + 𝑘𝑧)−1/𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆) 𝑧 1 + 𝑘𝑧 𝑘 = (1 + 𝑘𝑧)(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 + 𝑘𝑧)−1/𝑘 } 𝑟 ] 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟 ) 𝑛 𝑘=0 𝑘! 𝑆 𝑛, 𝑘 𝑀𝑞𝑘 𝜇,𝜈,𝛾,𝜉 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 𝑘 (4.6) (𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1 ; 𝑘 ≠ 0) Replacing 𝑧 by 𝑧 1−𝑘𝑧 and 𝑥 by 𝑥 (1−𝑘𝑧)1/𝑘the above equation immediately yields,
- 9. A Class of Polynomials Associated with Differential Operator and with a Generalization of Bessel … DOI: 10.9790/5728-11165563 www.iosrjournals.org 63 | Page ∞ 𝑘=0 𝑀𝑞𝑘 (𝜇,𝜈,𝛾,𝜉) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑧 𝑘 = (1 − 𝑘𝑧) −𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑧)− 1 𝑘 } 𝑟] 𝑛 𝑘=0 𝑘! 𝑆(𝑛, 𝑘)𝑀𝑞𝑘 𝜇,𝜈,𝛾,𝜉 x 1 − 𝑘𝑧 − 1 𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 1 − 𝑘𝑧 𝑘 (4.7) (𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1 ; 𝑘 ≠ 0) Similarly, theorem 4.1, applied to the generating function (2.6), would readily gives us the generating function ∞ 𝑘=0 𝑘 𝑛 𝑀𝑞𝑘 (𝜇,𝜈,𝛾,𝜉−𝑘𝑛) (𝑥(1 − 𝑘𝑧)1/𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆) 𝑧 1 − 𝑘𝑧 𝑘 = (1 − 𝑘𝑧)1−(𝜉+𝜆)/𝑘 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 − 𝑘𝑧)1/𝑘 } 𝑟 ] 𝐽𝜈,𝑞 𝜇,𝛾 (−𝑝𝑥 𝑟) 𝑛 𝑘=0 𝑘! 𝑆 𝑛, k 𝑀𝑞𝑘 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 𝑥; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 𝑘 (4.8) (𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1 ; 𝑘 ≠ 0) Thus for replacing𝑧 by 𝑧 1+𝑘𝑧 and 𝑥 by 𝑥 (1+𝑘𝑧)1/𝑘 the above equation immediately yields, ∞ 𝑘=0 𝑘 𝑛 𝑀𝑞𝑘 (𝜇,𝜈,𝛾,𝜉−𝑘𝑛) (𝑥; 𝑟, 𝑝, 𝑘, 𝜆)𝑧 𝑘 = (1 + 𝑘𝑧)−1+ 𝜉+𝜆 𝑘 𝐽𝜈,𝑞 𝜇,𝛾 −𝑝𝑥 𝑟 𝐽𝜈,𝑞 𝜇,𝛾 [−𝑝{𝑥(1 + 𝑘𝑧) 1 𝑘 } 𝑟] 𝑛 𝑘=0 𝑘! 𝑆 𝑛, 𝑘 𝑀𝑞𝑘 𝜇,𝜈,𝛾,𝜉−𝑘𝑛 x 1 + 𝑘𝑧 1 𝑘 ; 𝑟, 𝑝, 𝑘, 𝜆 𝑧 1 + 𝑘𝑧 𝑘 (4.9) (𝑛 ∈ 𝑁0;|𝑧| < |𝑘|−1 ; 𝑘 ≠ 0) References [1] H.M. Srivastava and J.P. Singhal , A class of polynomials defined by generalized Rodrigue's formula , Ann. Mat. Pura Appl. Ser. ,Ser. IV 90 (1971), 75-85 . [2] H.M. Srivastava, Some families of series transformations related to Euler-Knopp transformation, Internat. J. Nonlinear Sci. Numer. Simulation,2 (2001), 83-88. [3] K.Y. Chen, C.J. Chyan, and H.M. Srivastava, Some polynomials associated with a certain family of differential operator, Journal of Mathematical Analysis and Applications,268 (2002), 344-377. [4] A.K. Shukla and J.C. Prajapati, A general class of polynomials associated with generalized Mittag- Leffler function , Integral Transform and Special Function, Vol.19, No.1 (2008), 23-34. [5] A.K. Shukla, and J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl.,336 (2007), 797-811. [6] M. Singh, M.A. Khan, and A.H. Khan, On some properties of generalized Bessel-Maitland function, International Journal of Mathematics Trends and Technology, Volume 14, Number 1, Oct 2014. [7] K.R. Patil, and N.K. Thakre, Operational formulas for the function defined by a generalized Rodrigue's formula-II, Shivaji Univ. J.15 (1975), 1-10. [8] H.M. Srivastava, and H. L. Manocha, A Treatise on Generating Functions(Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984). [9] J.H. Hubble, and H.M. Srivastava, Certain theorem on bilateral generating functions involving Hermite, Laguerre and Gegenbauer polynomials, Journal of Mathematical Analysis and Applications, 152 (1990), 343-353. [10] H. Buchholz, The confluent hypergeometric function (Translated from the German by H. Lichtblau and K. Wetzel, Springer tract in Natural Philosophy, vol. 15, Springer-Verlag, New York, 1969). [11] J. Riodan, Combinatorial Identities, (New York/London/Sydney; Wiley, 1968). [12] H.M. Srivastava, Some families of generating functions associated with Stirling number of second kind, Journal of Mathematical Analysis and Applications,251 (2000), 752-759.