The Uniformization of Certain Algebraic Hypergeometric Functions

TL;DR

This paper explores the algebraic nature of certain hypergeometric functions, showing how they can be explicitly uniformized using roots of trinomials and analyzing the associated algebraic curves.

Contribution

It generalizes previous results by representing algebraic hypergeometric functions with finite but imprimitive monodromy as algebraic functions of trinomial roots, and provides explicit parametrizations.

Findings

Many algebraic hypergeometric functions can be expressed via roots of trinomials.

The Schwarz curve's genus determines the possibility of rational parametrization.

Explicit uniformizations are constructed for various algebraic hypergeometric functions.

Abstract

The hypergeometric functions ${}_nF_{n-1}$ are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic ${}_nF_{n-1}$'s, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic ${}_nF_{n-1}$'s. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic ${}_nF_{n-1}$'s, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.