Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 037, 40 pages      arXiv:2209.02227      https://doi.org/10.3842/SIGMA.2023.037

On $q$-Middle Convolution and $q$-Hypergeometric Equations

Yumi Arai and Kouichi Takemura
Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Received October 14, 2022, in final form May 19, 2023; Published online June 05, 2023

Abstract
The $q$-middle convolution was introduced by Sakai and Yamaguchi. In this paper, we reformulate $q$-integral transformations associated with the $q$-middle convolution. In particular, we discuss convergence of the $q$-integral transformations. As an application, we obtain $q$-integral representations of solutions to the variants of the $q$-hypergeometric equation by applying the $q$-middle convolution.

Key words: hypergeometric function; $q$-hypergeometric equation; middle convolution; $q$-integral.

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