Parametric transformations between the Heun and Gauss hypergeometric functions

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Vidunas, Raimundas
Filipuk, Galina
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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.
Comment: One-parameter transformations added; 35 pages, 6 tables
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Mathematics - Classical Analysis and ODEs, 33E30, 33C05
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