Polynomial solutions of $q$-Heun equation and ultradiscrete limit

TL;DR

This paper investigates polynomial solutions of the $q$-Heun equation, providing conditions for real, distinct roots of the spectral polynomial, and explores the ultradiscrete limit to analyze these roots and solutions.

Contribution

It introduces sufficient conditions for the roots of the spectral polynomial to be real and distinct, and applies the ultradiscrete limit to study the roots and zeros of polynomial solutions.

Findings

Roots of the spectral polynomial are real and distinct under certain conditions.

Ultradiscrete limit clarifies the structure of roots and zeros.

Provides a link between polynomial solutions and quasi-exact solvability.

Abstract

We study polynomial-type solutions of the $q$-Heun equation, which is related with quasi-exact solvability. The condition that the $q$-Heun equation has a non-zero polynomial-type solution is described by the roots of the spectral polynomial, whose variable is the accessory parameter $E$. We obtain sufficient conditions that the roots of the spectral polynomial are all real and distinct. We consider the ultradiscrete limit to clarify the roots of the spectral polynomial and the zeros of the polynomial-type solution of the $q$-Heun equation.