Expansion Formulas of Basic Hypergeometric Series via the (1-xy,y-x)--Inversion and Its Applications

TL;DR

This paper develops a new expansion formula for basic hypergeometric series using a specific matrix inversion, leading to generalizations of known $q$-series identities and new transformation formulas.

Contribution

It introduces an $(1-xy,y-x)$-expansion formula for hypergeometric series based on a novel matrix inversion approach, generalizing many existing $q$-series expansions.

Findings

Derived new transformation formulas for $q$-series

Provided a novel approach to Askey-Wilson polynomials

Extended Ramanujan's ${}_1 ext{ extsterling}_1$ summation formula

Abstract

With the use of the $(f,g)$-matrix inversion under specializations that $f=1-xy,g=y-x$, we establish an $(1-xy,y-x)$-expansion formula. When specialized to basic hypergeometric series, this $(1-xy,y-x)$-expansion formula leads us to some expansion formulas expressing any ${}_{r}\phi_{s}$ series in variable $x~t$ in terms of a linear combination of ${}_{r+2}\phi_{s+1}$ series in $t$, as well as various specifications. All these results can be regarded as common generalizations of many konwn expansion formulas in the setting of $q$-series. As specific applications, some new transformation formulas of $q$-series including new approach to the Askey-Wilson polynomials, the Rogers-Fine identity, Andrews' four-parametric reciprocity theorem and Ramanujan's ${}_1\psi_1$ summation formula, as well as a transformation for certain well-poised Bailey pairs, are presented.