On an expansion formula of Liu

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' ${}_6{\varphi }_5$ summation, an $A_n$ extension of Sylvester's identity, some multiple expansion formulas for $(q)_{\infty}^{m},\text{\ensuremath{π_{q}}}$ 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 ${}_6{\varphi }_5$ summation, an $A_n$ extension of a generalization of Fang's identity and an $A_n$ extension of Andrews' expansion formula.

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