Transformations of Heun's equation and its integral relations

TL;DR

This paper explores variable transformations that preserve the form of Heun's equation kernels, leading to new hypergeometric-based kernels for Heun and confluent Heun equations, expanding solution methods.

Contribution

It introduces novel transformations that generate new kernels for Heun's and confluent Heun equations, including Lambe-Ward-type and Erdélyi-type kernels, through a systematic approach.

Findings

New kernels for Heun's equation derived from hypergeometric functions.

Transformation methods produce kernels for confluent Heun equations.

Provides a framework for generating integral relations among solutions.

Abstract

We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting process, also afford new kernels for the confluent Heun equation.