## Advanced Studies in Pure Mathematics

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

Yasushi Kajihara

#### 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.

#### Article information

Dates
Revised: 7 July 2016
First available in Project Euclid: 21 September 2018

https://projecteuclid.org/ euclid.aspm/1537499428

Digital Object Identifier
doi:10.2969/aspm/07610247

Mathematical Reviews number (MathSciNet)
MR3837924

Zentralblatt MATH identifier
07039305

#### Citation

Kajihara, Yasushi. A unified approach to transformations for multiple basic hypergeometric series of type $A$. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 247--274, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610247. https://projecteuclid.org/euclid.aspm/1537499428