Parametric transformations between the Heun and Gauss hypergeometric functions

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.

One-parameter transformations added; 35 pages, 6 tables

Cited by in corpus (13)

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• The third exactly solvable hypergeometric quantum-mechanical potential
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• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Generalized hypergeometric solutions of the Heun equation
• Heun functions and diagonals of rational functions (unabridged version)
• On the derivatives of the Heun functions
• Solutions of Heun's general equation and elliptic Darboux equation
• Integrability of the Trapped Ionic System
• A new exactly integrable hypergeometric potential for the Schrödinger equation
• The third five-parametric hypergeometric quantum-mechanical potential