Parametric transformations between the Heun and Gauss hypergeometric functions

TL;DR

This paper catalogs 61 parametric transformations of maximal degree 12 that convert hypergeometric functions into Heun functions and vice versa, enhancing understanding of their interrelations through explicit pull-back formulas.

Contribution

It provides a comprehensive list of hypergeometric-to-Heun transformations with a free parameter, including explicit reduction formulas and transformation degrees.

Findings

61 parametric transformations identified

Transformations include maximal degree 12

Most transformations illustrated by reduction formulas

Abstract

The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.