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

SIGMA 20 (2024), 083, 22 pages      arXiv:2309.09341      https://doi.org/10.3842/SIGMA.2024.083

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

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

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.

Key words: kernel function; Jackson integral; Heun equation; $q$-Heun equation; Ruijsenaars system.

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