A General Family of $q$-Hypergeometric Polynomials and Associated Generating Functions

TL;DR

This paper introduces a broad family of $q$-hypergeometric polynomials, explores their generating functions and identities, and connects these results with existing literature, highlighting the redundancy of $(p,q)$-variations.

Contribution

It presents a new general family of $q$-hypergeometric polynomials and derives several novel generating function identities and transformational relations.

Findings

Extended generating functions for the new family

Bilinear generating functions of Srivastava-Agarwal type

Connections with previous $q$-series results

Abstract

In this paper, we introduce a general family of $q$-hypergeometric polynomials and investigate several $q$-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of $q$-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized $q$-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various $q$-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called $(p,q)$-variations of the $q$-results, which we have investigated here, because the additional parameter $p$ is obviously redundant.