On reducing the Heun equation to the hypergeometric equation

2002/03/31 by Robert S. Maier · 8 citations
Mathematics · Physics and Astronomy · #Nonlinear Waves and Solitons #math-ph #math.CA #math.MP #msc:33C05 #msc:33E30 #msc:34M35

paper · pdf · doi:10.1016/j.jde.2004.07.020

published as J. Differential Equations 213 (2005) 171-203 · 36 pages, a few additional misprints corrected

arxiv created 2004/08/23 · openalex publication_date 2004/09/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. [See K. Kuiken, "Heun's equation and the hypergeometric equation", SIAM Journal on Mathematical Analysis 10:3 (1979), 655-657.]

Citations

• Algebraic solutions of the Lamé equation, revisited

Cited by

• Renormalization, isogenies and rational symmetries of differential equations
• Heun equations coming from geometry
• A classification of coverings yielding Heun-to-hypergeometric reductions
• Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces
• Heun functions and combinatorial identities
• Analytic Results in the Position-Dependent Mass Schrödinger Problem
• The 192 solutions of the Heun equation
• HEUN FUNCTIONS VERSUS ELLIPTIC FUNCTIONS

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• A classification of coverings yielding Heun-to-hypergeometric reductions