# Heun's differential equation and its q-deformation

## TL;DR

This paper explores the q-deformation of Heun's differential equation, analyzing solutions and limits as q approaches 1 and 0, to understand its properties and connections to classical equations.

## Contribution

It introduces and investigates polynomial solutions of the q-Heun equation and examines its limits, connecting it to classical Heun's differential equation.

## Key findings

- Polynomial solutions of q-Heun equation identified
- Limit q→1 recovers classical Heun's equation
- Limit q→0 reveals ultradiscrete behavior

## Abstract

The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding to Heun's differential equation and the ultradiscrete limit $q\to 0$ are considered.