Kernel Function, $q$-Integral Transformation and $q$-Heun Equations

Kouichi Takemura

Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Abstract: We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the perspective of the $q$-integral transformation.

Keywords: kernel function, Jackson integral, Heun equation, $q$-Heun equation, Ruijsenaars system.

MSC: 39A13, 44A20

Received: February 14, 2024; in final form September 12, 2024; Published online September 19, 2024

Language: English

DOI: 10.3842/SIGMA.2024.083

ArXiv: 2309.09341