MR3114824 Raimundas VIDUNAS and Galina FILIPUK Raimundas VIDUNAS and Galina FILIPUK Parametric Transformations between the Heun and Gauss Hypergeometric Functions Funkcialaj Ekvacioj. Serio Internacia 56 2013 271--321 http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-2/56_271.pdf http://www.ams.org/mathscinet-getitem?mr=MR3114824 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 $\mathbb{P}^1\to\mathbb{P}^1$. 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. Heun equation, Hypergeometric equation, Heun function, Gauss hypergeometric function. Primary 33E30, Secondary 33C05. 56-271 2013 Parametric Transformations between the Heun and Gauss Hypergeometric Functions Raimundas VIDUNAS and Galina FILIPUK Raimundas VIDUNAS and Galina FILIPUK 1 Andrews, G. E.; Askey, R.; Roy, R. Special Functions Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge 1999 MR1688958
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