This article is cited in 14 papers

On $q$-Deformations of the Heun Equation

Kouichi Takemura

Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan

Abstract: The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars–van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator.

Keywords: Heun equation; $q$-deformation; regular singularity; quasi-exact solvability; degeneration.

MSC: 39A13; 33E10

Received: January 18, 2018; in final form May 29, 2018; Published online June 18, 2018

Language: English

DOI: 10.3842/SIGMA.2018.061

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