On $q$-deformations of the Heun equation

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.

References in corpus (1)

• Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations

Cited by in corpus (14)

• Algebraic Heun operator and band-time limiting
• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• Heun algebras of Lie type
• The Heun-Racah and Heun-Bannai-Ito algebras
• Quantum Representation of Affine Weyl Groups and Associated Quantum Curves
• The rational Heun operator and Wilson biorthogonal functions
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Variants of $q$-hypergeometric equation
• Variants of confluent q-hypergeometric equations
• $q$-Middle Convolution and $q$-Painlevé Equation
• Polynomial solutions of $q$-Heun equation and ultradiscrete limit
• Heun's differential equation and its q-deformation
• q-Heun equation and initial-value space of q-Painlevé equation