# New Generating Function Relations for theq-Generalized Cesàro Polynomials

Authors: Nejla Özmen
Journal: Journal of Function Spaces
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

The Cesàro polynomialsgn(s)(x) are defined by the generating relation ([1], p. 449, Problem 20):(1)∑n=0∞gnsxtn=1-t-s-11-xt-1.It is from (1) that(2)gnsx=s+nnF21-n,1;-s-n;x,where F12 denotes Gauss’s hypergeometric series.Lin et al. [2] introduced the generalized Cesàro polynomials as follows:(3)gnsλ,x=s+nnF12-n,λ;-s-n;x.It is noted that the special case λ=1 of (3) reduces immediately to the Cesàro polynomials defined by (2).Furthermore, they satisfy the generating functions [3]:(4)∑n=0∞gnsλ,xtn=1-t-s-11-xt-λand(5)∑n=0∞n+mngn+msλ,xtn=1-t-s-m-11-xt-λgmsλ,x1-t1-xt,where m=0,1,2,…..The purpose of this study is to obtainq-analogue of generalized Cesàro polynomials as q-analogue of the production functions mentioned above. The structure of this paper is as follows.In Section2, we give some preliminaries on q-calculus. In Section 3, we define some q-analogue of Cesàro polynomials. In Section 4, theorems are given for bilinear and bilateral generating functions for q-generalized Cesàro polynomials. In Section 5, the application of the theorems given in Section 4 will be given.


## 2. Someq-Calculus: The Definitions

Letq∈C,0<q<1. A q-analogue of the hypergeometric series pFr is the basic hypergeometric series [4]:(6)ϕrsα1,…,αr;q,zβ1,…,βs=∑k=0∞α1;qk⋯αr;qkβ1;qk⋯βs;qk-1kqkk-1/21+r+szkq;qk,where q≠0 when r>s+1, and (βi) are such that the denominator never vanishes. We also need to define some other q-analogues, such as the q-analogue of a number αq, factorial αq!, and the Pochhammer symbol (rising factorial) αn. These q-analogues are given as follows:(7)αq=1-qα1-q,0q!=1,nq!=∏k=1nkq,n∈N.The number(μ;q)ω is given by(8)μ;qω≔μ;q∞μqω;q∞,where(9)μ;q∞≔∏s=0∞1-μqsand μ,ω are arbitrary parameters so that (10)μ;qω≔1ifω=01-μ1-μq⋯1-μqω-1ifω=1,2,3,…;see, for instance, [5], pp. 413-414.Theq-gamma function [4] is defined by(11)Γqx=q;q∞qx;q∞1-q1-x,x∈C,x∉0,-1,-2,….And it satisfies(12)Γqx+1=1-qx1-qΓqx.Definition 1.
Theq-analogue of Cesàro’s polynomial is defined as follows [6]:(13)gnsx;q=q1+s;qnq;qnϕ21q-n,q;q,xq-s-n=∑k=0nk+ssqxqsn-k,where ϕ21 denotes q-hypergeometric function and defined by [6](14)ϕ21q-n,q;q,xq-s-n=∑k=0nq-n;qkq;qkq-s-n;qkxkq;qk.Definition 2.
Theq-Cesàro polynomials satisfy the following generating function [6, 7]:(15)∑n=0∞gnsx;qtn=11-qsxtt;qs+1.Following Saxena, Modi, and Kalla [8], the basic analogue of the Fox’s H-function is defined as(16)HP,QM,Na,αx;qb,β=12πi∫CΦs;qxsdqs,where(17)Φs;q=∏j=1MGqbj-βjs∏j=1NGq1-aj+αjsπ∏j=M+1QGq1-bj+βjs∏j=N+1PGqaj-αjsGq1-ssin⁡πsand(18)Gqs=∏n=0∞1-qa+n-1=1qa;q∞.Also 0≤M≤Q,0≤N≤P,αi’s and βj’s are positive integers. The contour C is a line parallel to Re⁡(ws)=0 with indentations if necessary, in such a manner that all the poles of G(qbj-βjs), 1≤j≤M are to the right and those of G(q1-aj+αjs),1≤j≤N are to the left of C. For large values of s, the integral converges if Re⁡slog⁡(x)-log⁡sin⁡πs<0 on the contour C, i.e., if arg⁡(x)-w2w1-1log⁡x<π, where 0<q<1,log⁡q=-w=-(w1+iw2),w1 and w2 are real.Further, if we setαi=βj=1,∀i and j in (16), we obtain the basic analogue of Meijer’s G−function due to Saxena, Modi, and Kalla [8]:(19)GP,QM,Na1,…,aPx;qb1,…,bQ=12πi∫CΦ′s;qxsdqs,where(20)Φ′s;q=∏j=1MGqbj-s∏j=1NGq1-aj+sπ∏j=M+1QGq1-bj+s∏j=N+1PGqaj-sGq1-ssin⁡πs.Detailed account of Meijer’sG-function, Fox’s H-function, and various functions expressed by Fox’s H-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 Gq(·) 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 q-calculus) can be found on the theory of approaches of operators [15, 16].


## 3. Construction of theq-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 q-generalized Cesàro polynomials gn(s)(λ,x;q) given by (3).Definition 3.
Theq-generalized Cesàro polynomials gn(s)(λ,x;q) given by (3) are written as follows:(21)gnsλ,x;q≔qs+1;qnq;qnϕ21q-n,qλ;q,xq-s-n=∑k=0nn-k+sn-kqqλ;qkxqskq;qk.It is noted that the special caseλ=1 of (21) reduces immediately to the generalized Cesàro polynomials defined by (4).Theorem 4.
Theq-generalized Cesàro polynomials have the following generating function relation:(22)1qsxt;qλt;qs+1=∑k=0∞gnsλ,x;qtn,where t<x1-1,k∈N0.Proof.
Using the well-knownq-binomial theorem (see [19], p. 241-248, [5], p. 416) and from (21), we get(23)∑n=0∞gnsλ,x;qtn=∑n=0∞qs+1;qnq;qnϕ21q-n,qλ;q,xq-s-ntn=∑n=0∞qs+1;qnq;qn∑k=0nq-n;qkqλ;qkq;qkq-s-n;qkxktn.Now making use of the identity(24)q-n;qk=q;qnq;qn-k-1kqk2-nk,we have(25)∑n=0∞gnsλ,x;qtn=∑n=0∞qs+1;qnq;qn∑k=0nq;qn-1kqk2-nkq;qs+n-kqλ;qkq;qn-kq;qs+n-1kqk2-nk-skq;qkxktn=∑n=0∞∑k=0nqs+1;qnq;qs+n-kqλ;qkqskq;qn-kq;qs+nq;qkxktn=∑n=0∞∑k=0∞qs+1;qn+kq;qs+nqλ;qkqskq;qnq;qs+n+kq;qkxktn+k=∑n=0∞qs+1;qnq;qn∑k=0∞qλ;qkqskq;qkxktn+k=∑n=0∞qs+1;qnq;qntn∑k=0∞qλ;qkq;qkqsxtk=1-qs+1t∞1-t∞1-qλ+sxt∞1-qsxt∞=1t;qs+1qsxt;qλ,which completes the proof.


## 4. Theq-Generating Relations

In this section, we have obtained bilinear and bilateral generating functions of various families for theq-analogue of the generalized Cesàro polynomials gn,q(s)(λ,x) given by (22). In addition, we will get a specific linear q-generating relationship that includes the basic analogue of Fox’s H-function and a general class of q-hypergeometric polynomials. We begin by stating the following theorem.Theorem 5.
For nonvanishing functionΩμ(y1,…,ys) of complex variables y1,…,ys(s∈N) and of complex order μ, let (26)Λμ,ψy1,…,ys;ζ≔∑k=0∞akΩμ+ψky1,…,ysζk,ak≠0,μ,ψ∈Cand(27)Θn,pμ,ψλ,x;q;y1,…,ys;ξ≔∑k=0n/pakgn-pksλ,x;qΩμ+ψky1,…,ysξk,where [n/p] denotes the integer part of n, p∈R.
Then,(28)∑n=0∞Θn,pμ,ψλ,x;q;y1,…,ys;ηtptn=1qsxt;qλt;qs+1Λμ,ψy1,…,ys;η.Proof.
LetS denote the first member of the assertion (28) of Theorem 5. Taking ξ→η/tp and sum from n=0 to ∞ and also multiplying by tn, we have(29)S=∑n=0∞Θn,pμ,ψλ,x;q;y1,…,ys;ηtptn=∑n=0∞∑k=0n/pakgn-pksλ,x;qΩμ+ψky1,…,ysηktn-pk.Replacing n by n+pk, we can write (30)S=∑n=0∞∑k=0∞akgnsλ,x;qΩμ+ψky1,…,ysηktn=∑n=0∞gnsλ,x;qtn∑k=0∞akΩμ+ψky1,…,ysηk=1qsxt;qλt;qs+1Λμ,ψy1,…,ys;η.which completes the proof.Theorem 6.
LetSn,qn=0∞ be an arbitrary bounded sequence, let M,N,P,Q be positive integers such that 0≤M≤Q,0≤N≤P, let h>0, and let m be an arbitrary positive integer. Then the following bilateral q-generating relation holds:(31)∑n=0∞Sn,qgnsλ,ρx;qHP+1,QM,N+11-μ-n,h,a,αy;qb,βtn=12πi∫CΦu;qΓqμ+hu1-qμ+hu-1q;q∞×∑k=0∞∑n=0∞Sn+k,qqμ+hu;qkqμ+k+hu;qnqs+1;qnqλ;qkq;qnq;qkqsρxtkyutndqu,where 0<q<1 and ρ and μ are arbitrary numbers.Proof.
Denoting, for convenience, the left-hand side of (31) by L and using the contour integral representation (16) for the q-analogue of Fox’s H-function and the definition (21) for the q-generalized Cesàro polynomials, we get(32)L=12πi∑n=0∞Sn,qqs+1;qnq;qnϕ21q-n,qλ;q,ρxq-s-n·∫CΦu;qGqμ+n+huyudqutn.Changing the order of summations and integration, we obtain(33)L=12πi∫CΦu;q∑n=0∞∑k=0nSn,qGqμ+n+hulqs+1;qnq;qnq-n;qkqλ;qkq-s-n;qkq;qkρxkyutndqu,where Φ(u;q) is given by (17). Using of the relation for q-gamma function, namely, (34)Gqα=Γqα1-qα-1q;q∞,we obtain(35)L=12πi∫CΦu;qΓqμ+hu1-qμ+hu-1q;q∞·∑n=0∞∑k=0nSn,qqμ+hu;qnqs+1;qnq;qnq-n;qkqλ;qkq-s-n;qkq;qkρxkyutndqu.By using identity (24), we have(36)L=12πi∫CΦu;qΓqμ+hu1-qμ+hu-1q;q∞∑n=0∞∑k=0nSn,qqμ+hu;qnqs+1;qnqλ;qkq;qs+n-kq;qs+nq;qn-kq;qkqksρxkyutndqu.
Again, changing the order of summations and making use of the series rearrangement relation [1](37)∑n=0∞∑k=0nBk,n=∑k=0∞∑n=0∞Bk,n+k,we obtain(38)L=12πi∫CΦu;qΓqμ+hu1-qμ+hu-1q;q∞∑k=0∞∑n=0∞Sn+k,qqμ+hu;qn+kqs+1;qn+kqλ;qkq;qs+nq;qs+n+kq;qnq;qkqksρxkyutn+kdqu.Now by interchanging the order of contour integral and summation, and using the q-identities [4], namely,(39)α;qn+k=α;qnαq;qkand(40)α;qn=Γα+n1-qnΓαn>0,we obtain(41)L=12πi∫CΦu;qΓqμ+hu1-qμ+hu-1q;q∞∑k=0∞∑n=0∞Sn+k,qqμ+hu;qkqμ+k+hu;qnqs+1;qnqλ;qkq;qnq;qkqsρxtkyutndqu.


## 5. Special Cases

As an application of the above in Theorem5, when the multivariable function Ωμ+ψk(y1,…,yr),k∈N0,r∈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(42)r=1andΩμ+ψky=gμ+ψksλ,y;qin Theorem 5, where the q-generalized Cesàro polynomials are generated by (22). We thus led to the following result which provides a class of bilinear generating functions for the q-generalized Cesàro polynomials.Corollary 1.
If(43)Λμ,ψλ,y;q;ζ≔∑k=0∞akgμ+ψksλ,y;qζk,ak≠0,μ,ψ∈C,then we have(44)∑n=0∞∑k=0n/pakgn-pksλ,x;qgμ+ψksλ,y;qηktpktn=1qsxt;qλt;qs+1Λμ,ψλ,y;q;η.Remark 2.
Using the generating relation (22) for the generalized Cesàro polynomials and getting ak=1, μ=0, ψ=1 in Corollary 1, we find that (45)∑n=0∞∑k=0n/pgn-pksλ,x;qgksλ,y;qηktn-pk=1qsxt;qλt;qs+11qsyη;qλη;qs+1,where |t|<1,|η|<1.By assigning suitable special values to the sequenceSn,qn=0∞, our main result (Theorem 6) can be applied to derive certain bilateral q-generating relations for the product of orthogonal q-polynomials and the basic analogue of Fox’s H-function. To illustrate this, we consider the following example.Setρ=1 and (46)Sn,q=1qλ;qkqs+1;qn-k.Thus, in view of the above relations, Theorem6 yields the q-generating relation involving q-generalized Cesàro polynomial and the basic Fox’s H-function as below.Corollary 3.
The following bilateral generating function holds true:(47)∑n=0∞1qλ;qkqs+1;qn-kgnsλ,ρx;qHP+1,QM,N+11-μ-n,h,a,αy;qb,βtn=11-tμ∑k=0∞qsxtktqμ;qkq;qk12πi∫CΦu;qΓqμ+k+hu1-qμ+k+hu-1tqμ+k;qhuq;q∞yudqu=11-tμ∑k=0∞qsxtktqμ;qkq;qkHP+1,QM,N+1y1-tqμ+kh;q1-μ-k,h,a,αb,β,where t<1,0<q<1 and μ is arbitrary numbers.If we takeαi=βj=1 for all i and j and m=h=1 and set (19) and(48)Sn,q=1qλ;qkqs+1;qn-k,in Theorem 6, we have the following bilateral generating functions for the q-generalized Cesàro polynomials.Corollary 4.
Let{Sn,q}n=0∞ be an arbitrary bounded sequence and let M,N,P,Q be positive integers satisfying 0≤M≤Q,0≤N≤P. Then the following bilateral q-generating relation for the function Gq(·) holds:(49)∑n=0∞1qλ;qkqs+1;qn-kgnsλ,ρx;qGP+1,QM,N+11-μ-n,a1,…,aPy;qb1,…,bQtn=11-tμ∑k=0∞qsρxtktqμ;qkq;qkGP+1,QM,N+11-μ-k,a1,…,aPy1-tqμ+kh;qb1,…,bQ,where t<1,0<q<1 and ρ and μ are arbitrary numbers.For every suitable choice of the coefficientsak(k∈N0), if the multivariable function Ωμ(y1,…,ys)(s=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 q-generalized Cesàro polynomials gn(s)(λ,x;q) defined by (22).We conclude with the remark that by suitably assigning values to the sequence{Sn,q}n=0∞, the q-generating relation (31), being of general nature, will lead to several generating relations for the product of orthogonal q-polynomials and the basic analogue of the Fox’s H- functions.



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