Transformations of Heun's equation and its integral relations

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.