A unified approach to transformations for multiple basic hypergeometric series of type $A$

Abstract

We propose a unified approach to transformation and summation formulas for multiple basic hypergeometric series of type $A$ on the basis of balanced duality transformations. We study two classes of transformations, one between an $n$-ple sum and an $m$-ple sum, and the other between two $n$-ple sums. Though the latter are not simple special cases of the former, we can still derive them from the former in a systematic way. Our derivation utilizes the fact that some multiple basic hypergeometric series have hidden symmetry which originates from the relation to basic hypergeometric series in one variable. We also give remarks on the related summation formulas.