Elementary derivations of summations and transformation formulas for q-series

TL;DR

This paper offers elementary, simplified derivations of summation and transformation formulas for q-series, providing new insights and a novel family of formulas for well poised basic hypergeometric series, enhancing understanding of q-series theory.

Contribution

It introduces elementary derivation methods for q-series formulas, including a new family of summation formulas for well poised basic hypergeometric series.

Findings

Elementary derivations of q-series formulas

Simpler proofs compared to classical texts

New summation formulas for _{6+2k}W_{5+2k} series

Abstract

We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric Series" book (which we will refer to as BHS), the Bailey [1935] and Slater [1966] books, and in some papers; thus providing deeper insights into the theory of q-series. Our main emphasis is on methods that can be used to derive formulas, rather than to just verify previously derived or conjectured formulas. In section 5 this approach leads to the derivation of a new family of summation formulas for very well poised basic hypergeometric series _{6+2k}W_{5+2k}, k = 1,2,.... Several of the observations in this paper were presented, along with related exercises, in the author's minicourse on "q-Series" at the Fields Institute miniprogram on "Special functions, q-Series and Related Topics," June 12-14, 1995.