New Generating Function Relations for the q − Generalized

Copyright © 2019 Nejla Özmen.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The main purpose of this paper is to examine a basic (or q−) analogue of the generalized Cesàro polynomials described here. We derive a bilateral q−generating function involving basic analogue of Fox’sH−function and q−generalized Cesàro polynomials.

where  ̸ = 0 when  >  + 1, and (  ) are such that the denominator never vanishes.We also need to define some other −analogues, such as the −analogue of a number []  , factorial []  !, and the Pochhammer symbol (rising factorial) ()  .These −analogues are given as follows: The number (; )  is given by where and ,  are arbitrary parameters so that see, for instance, [5], pp.413-414.The −gamma function [4] is defined by And it satisfies Definition 1.The −analogue of Cesàro's polynomial is defined as follows [6]: where 2  1 denotes −hypergeometric function and defined by [6] 2 Definition 2. The −Cesàro polynomials satisfy the following generating function [6,7]: Following Saxena, Modi, and Kalla [8], the basic analogue of the Fox's −function is defined as where and Also 0 ≤  ≤ , 0 ≤  ≤ ,   's and   's are positive integers.The contour  is a line parallel to Re() = 0 with indentations if necessary, in such a manner that all the poles of (   −   ), 1 ≤  ≤  are to the right and those of ( where Detailed account of Meijer's −function, Fox's −function, and various functions expressed by Fox's −function can be found in the research monographs of Mathai and Saxena [9,10], Srivastava, Gupta, and Goyal [11], and Mathai, Saxena, and Haubold [12].In addition, the basic functions of a variable that can be expressed in terms of (⋅) functions can be found in the works of Yadav and Purohit [13,14].In the last quarter of the twentieth century, the quantum calculus (also known as −calculus) can be found on the theory of approaches of operators [15,16].

The 𝑞−Generating Relations
In this section, we have obtained bilinear and bilateral generating functions of various families for the −analogue of the generalized Cesàro polynomials  () , (, ) given by ( 22).In addition, we will get a specific linear −generating relationship that includes the basic analogue of Fox's −function and a general class of −hypergeometric polynomials.We begin by stating the following theorem.
Theorem 6.Let { , } ∞ =0 be an arbitrary bounded sequence, let , , ,  be positive integers such that 0 ≤  ≤ , 0 ≤  ≤ , let ℎ > 0, and let  be an arbitrary positive integer.Then the following bilateral −generating relation holds: where Φ(; ) is given by (17).Using of the relation for −gamma function, namely, we obtain (36) Again, changing the order of summations and making use of the series rearrangement relation [1] we obtain Now by interchanging the order of contour integral and summation, and using the −identities [4], namely, and

Special Cases
As an application of the above in Theorem 5, when the multivariable function Ω + ( 1 , . . .,   ),  ∈ N 0 ,  ∈ N, is expressed in terms of simpler functions of one and more variables, then we can give additional applications of the above theorem.We first set and Ω + () = where || < 1, || < 1.
By assigning suitable special values to the sequence { , } ∞ =0 , our main result (Theorem 6) can be applied to derive certain bilateral −generating relations for the product of orthogonal −polynomials and the basic analogue of Fox's −function.To illustrate this, we consider the following example.
If we take   =   = 1 for all  and  and  = ℎ = 1 and set (19) and in Theorem 6, we have the following bilateral generating functions for the −generalized Cesàro polynomials.
For every suitable choice of the coefficients   ( ∈ N 0 ), if the multivariable function Ω  ( 1 , . . .,   ) ( = 2, 3, . ..) is expressed as an appropriate product of several simpler functions, the assertion of the above Theorem 5 can be applied in order to derive various families of multilinear and multilateral generating functions for the −generalized Cesàro polynomials  ()   (, ; ) defined by ( 22).We conclude with the remark that by suitably assigning values to the sequence { , } ∞ =0 , the −generating relation (31), being of general nature, will lead to several generating relations for the product of orthogonal −polynomials and the basic analogue of the Fox's functions.
(16)ting, for convenience, the left-hand side of (31) by  and using the contour integral representation(16)for the −analogue of Fox's −function and the definition (21) for the −generalized Cesàro polynomials, we get      }   .