Raimundas Vidunas

Transformations of some Gauss hypergeometric functions

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/l, 1/m such that k, l, m are positive integers and 1/k + 1/l + 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.

A classification of coverings yielding Heun-to-hypergeometric reductions

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). The same Belyi coverings appear in several other contexts. For example, 38 of the coverings appear in Herfutners list of elliptic surfaces over P1 with four singular fibers, as their j-invariants. In passing, we demonstrate an elegant way to show that there are no coverings P1-to-P1 with some branching patterns.

Parametric Transformations between the Heun and Gauss Hypergeometric Functions

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.

DEGENERATE GAUSS HYPERGEOMETRIC FUNCTIONS

In this paper we study terminating and ill-defined Gauss hypergeometric functions. For their hypergeometric equations, the set of 24 Kummer′s solutions degenerates. We describe those solutions and relations between them.

EXPRESSIONS FOR VALUES OF THE GAMMA FUNCTION

This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of

Computation of highly ramified coverings

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions

and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to evaluate, for example, algebraic Gauss hypergeometric functions by the Gauss identity. Also, algebraic independence of gamma values and their relation to the elliptic K-function are briefly discussed.

TRANSFORMATIONS AND INVARIANTS FOR DIHEDRAL GAUSS HYPERGEOMETRIC FUNCTIONS

An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation.

Parabolic cylinder functions: examples of error bounds for asymptotic expansions

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.

A generalization of Clausen’s identity

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 Clausens formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Kleins pullback transformations for the equations with a finite (dihedral) monodromy group.