Research Article | Open Access

# New Generating Function Relations for the Generalized Cesàro Polynomials

**Academic Editor:**Tuncer Acar

#### Abstract

The main purpose of this paper is to examine a basic (or ) analogue of the generalized Cesàro polynomials described here. We derive a bilateral generating function involving basic analogue of Fox’s function and generalized Cesàro polynomials.

#### 1. Introduction

The Cesàro polynomials are defined by the generating relation ([1], p. 449, Problem 20):It is from (1) thatwhere denotes Gauss’s hypergeometric series.

Lin et al. [2] introduced the generalized Cesàro polynomials as follows:It is noted that the special case of (3) reduces immediately to the Cesàro polynomials defined by (2).

Furthermore, they satisfy the generating functions [3]:andwhere .

The purpose of this study is to obtain analogue of generalized Cesàro polynomials as analogue of the production functions mentioned above. The structure of this paper is as follows.

In Section 2, we give some preliminaries on calculus. In Section 3, we define some analogue of Cesàro polynomials. In Section 4, theorems are given for bilinear and bilateral generating functions for generalized Cesàro polynomials. In Section 5, the application of the theorems given in Section 4 will be given.

#### 2. Some Calculus: The Definitions

Let . A analogue of the hypergeometric series is the basic hypergeometric series [4]:where when , 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 bywhereand are arbitrary parameters so that see, for instance, [5], pp. 413-414.

The gamma function [4] is defined byAnd it satisfies

*Definition 1. *The analogue of Cesàro’s polynomial is defined as follows [6]:where denotes hypergeometric function and defined by [6]

*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 aswhereandAlso ’s and ’s are positive integers. The contour is a line parallel to with indentations if necessary, in such a manner that all the poles of , are to the right and those of are to the left of . For large values of , the integral converges if on the contour , i.e., if , where and are real.

Further, if we set and in (16), we obtain the basic analogue of Meijer’s G−function due to Saxena, Modi, and Kalla [8]: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].

#### 3. Construction of the Generalized Cesàro Polynomials

In this section, with the help of the similar method as considered in [2, 5, 17, 18], we form the analogue of generalized Cesàro polynomials given by (3).

*Definition 3. *The generalized Cesàro polynomials given by (3) are written as follows:

It is noted that the special case of (21) reduces immediately to the generalized Cesàro polynomials defined by (4).

Theorem 4. *The generalized Cesàro polynomials have the following generating function relation:where .*

*Proof. *Using the well-known binomial theorem (see [19], p. 241-248, [5], p. 416) and from (21), we getNow making use of the identitywe havewhich completes the proof.

#### 4. The 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 5. *For nonvanishing function of complex variables and of complex order , let andwhere denotes the integer part of , **Then,*

*Proof. *Let denote the first member of the assertion (28) of Theorem 5. Taking and sum from to and also multiplying by , we haveReplacing by , we can write which completes the proof.

Theorem 6. *Let be an arbitrary bounded sequence, let be positive integers such that , let , and let be an arbitrary positive integer. Then the following bilateral generating relation holds:where and and are arbitrary numbers.*

*Proof. *Denoting, 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 getChanging the order of summations and integration, we obtainwhere is given by (17). Using of the relation for gamma function, namely, we obtainBy using identity (24), we haveAgain, changing the order of summations and making use of the series rearrangement relation [1]we obtainNow by interchanging the order of contour integral and summation, and using the identities [4], namely,andwe obtain

#### 5. Special Cases

As an application of the above in Theorem 5, when the multivariable function , is expressed in terms of simpler functions of one and more variables, then we can give additional applications of the above theorem.

We first setin Theorem 5, where the generalized Cesàro polynomials are generated by (22). We thus led to the following result which provides a class of bilinear generating functions for the generalized Cesàro polynomials.

Corollary 1. *Ifthen we have*

*Remark 2. *Using the generating relation (22) for the generalized Cesàro polynomials and getting , , in Corollary 1, we find that where .

By assigning suitable special values to the sequence , 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.

Set and

Thus, in view of the above relations, Theorem 6 yields the generating relation involving generalized Cesàro polynomial and the basic Fox’s function as below.

Corollary 3. *The following bilateral generating function holds true:where and is arbitrary numbers.*

If we take for all and and and set (19) andin Theorem 6, we have the following bilateral generating functions for the generalized Cesàro polynomials.

Corollary 4. *Let be an arbitrary bounded sequence and let be positive integers satisfying . Then the following bilateral generating relation for the function holds:where and and are arbitrary numbers.*

For every suitable choice of the coefficients , if the multivariable function 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 , 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.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that he has no conflicts of interest.

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#### Copyright

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.