Generating functions for some families of the generalized Al-Salam\u2013Carlitz q-polynomials

Hari Mohan Srivastava,Sama Arjika

doi:10.1186/s13662-020-02963-9

Abstract

In this paper, by making use of the familiar q-difference operators D_{q} and D_{q^{-1}}, we first introduce two homogeneous q-difference operators mathbb{T}(mathbf{a},mathbf{b},cD_{q}) and mathbb{E}(mathbf{a},mathbf{b}, cD_{q^{-1}}), which turn out to be suitable for dealing with the families of the generalized Al-Salam–Carlitz q-polynomials phi_{n}^{(mathbf{a},mathbf{b})}(x,y|q) and psi_{n}^{(mathbf{a},mathbf{b})}(x,y|q). We then apply each of these two homogeneous q-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas, and the Srivastava–Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam–Carlitz q-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.

Highlights

1 Introduction, definitions, and preliminaries The quantum polynomials constitute a very interesting set of special functions and orthogonal polynomials
The main objective of this paper is to investigate two families of the generalized Al-Salam–Carlitz q-polynomials φn(a,b)(x, y|q) and ψn(a,b)(x, y|q) by first representing them by the homogeneous q-difference operators T(a, b, cDq) and E(a, b, cDq–1 ), which we have introduced here
For the convenience of the reader, we provide a summary of mathematical notations and definitions, basic properties, and other relations to be used in the sequel

Summary

Introduction

1 Introduction, definitions, and preliminaries The quantum (or q-) polynomials constitute a very interesting set of special functions and orthogonal polynomials. Their generating functions appear in several branches of mathematics and physics (see, for details, [1,2,3,4,5]) such as (for example) continued fractions, Eulerian series, theta functions, elliptic functions, quantum groups and algebras, discrete mathematics (including combinatorics and graph theory), coding theory, and so on. For the convenience of the reader, we provide a summary of mathematical notations and definitions, basic properties, and other relations to be used in the sequel.

Objectives

Conclusion