On an expansion formula of Liu

TL;DR

This paper generalizes Liu's expansion formula to multiple basic hypergeometric series over the root system $A_{n}$, leading to new identities and extensions relevant to special functions and number theory.

Contribution

The paper introduces a comprehensive multiple expansion formula over the root system $A_{n}$ and derives several new extensions and identities, broadening Liu's original work.

Findings

Derived an $A_{n}$ Rogers' $_{6}\phi_{5}$ summation

Extended Sylvester's identity to the $A_{n}$ setting

Provided multiple expansion formulas for $(q)_{\infty}^{m}$ and related functions

Abstract

In this paper, we extend an expansion formula of Liu to multiple basic hypergeometric series over the root system $A_{n}.$ The usefulness of Liu's expansion formula in special functions and number theory has been shown by Liu and many others. We first establish a very general multiple expansion formula over the root system $A_{n}$ and then deduce several $A_{n}$ extensions of Liu's expansion formula. From these multiple formulas, we derive two groups of multiple expansion formulas for infinite products. As applications, we deduce an $A_{n}$ Rogers' $\text{}_{6}\phi_{5}$ summation, an $A_{n}$ extension of Sylvester's identity, some multiple expansion formulas for $(q)_{\infty}^{m},\text{\ensuremath{\pi_{q}}}$ and $1/\pi_{q}$, two $A_{n}$ extensions of the Rogers-Fine identity, an $A_{n}$ extension of Liu's extension of Rogers' non-terminating $_{6}\phi_{5}$ summation, an $A_{n}$ extension of a generalization of Fang's identity and an $A_{n}$ extension of Andrews' expansion formula.