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

SIGMA 14 (2018), 061, 16 pages      arXiv:1712.09564      https://doi.org/10.3842/SIGMA.2018.061
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### 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

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

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.

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

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