Algorithms and differential relations for Belyi functions

2016

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k, 1/ , 1/m that satisfy k, , m ∈ N and the hyperbolic condition 1/k + 1/ + 1/m < 1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.

Journal of Algebra, 2015

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k, 1/ , 1/m that satisfy k, , m ∈ N and the hyperbolic condition 1/k + 1/ + 1/m < 1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.

2005

Algebraic Gauss hypergeometric functions can be expressed explicitly in several ways. One attractive way is to pull-back their hypergeometric equations (with a finite monodromy) to Fuchsian equations with a finite cyclic monodromy, and express the algebraic solutions as radical functions on the covering curve. This article presents these pull-back transformations of minimal degree for the hypergeometric equations with the tetrahedral, octahedral or icosahedral projective monodromy. The minimal degree is 4, 6 or 12, respectively. The covering curves are called Darboux curves, and they have genus 0 or (for some icosahedral Schwarz types) genus 1.

Journal of Computational and Applied Mathematics, 2005

This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1/k, 1/ , 1/m such that k, , m are positive integers and 1/k + 1/ + 1/m < 1. All algebraic transformations of these Gauss hypergeometric functions are considered. We show that apart from classical transformations of degree 2, 3, 4, 6 there are several other transformations of degree 6, 8, 9, 10, 12, 18, 24. Besides, we present an algorithm to compute relevant Belyi functions explicitly.

2008

A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein's pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed from a data base (covering the Schwarz table) and using contiguous relations. Klein's pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group.

Integral Transforms and Special Functions

Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group is much simpler. This article considers the classical 3F2-functions with the projective monodromy group PSL(2, F7) and their pull-back transformations of degree 21 that reduce the projective monodromy to the dihedral group D4 of 8 elements.

Bulletin of the London Mathematical Society, 1996

arXiv: Classical Analysis and ODEs, 2018

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations were already considered for algebraic 2F1-functions. This article presents Darboux evaluations for the classical case of 3F2-functions with the projective monodromy group PSL(2,F7). As an application, appealing modular evaluations of the same 3F2-functions are derived.

2008

Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered.

Kyushu Journal of Mathematics, 2012

Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Pull-back transformations for the dihedral hypergeometric equations are also presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group.