The uniformization of certain algebraic hypergeometric functions

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The hypergeometric functions Fn-1n 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 Fn-1n'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 Fn-1n'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 Fn-1n's, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

Original languageEnglish (US)
Pages (from-to)86-138
Number of pages53
JournalAdvances in Mathematics
Volume253
DOIs
StatePublished - Mar 1 2014

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Algebraic function
Uniformization
Hypergeometric Functions
Parametrization
Monodromy
Curve
Genus
Binomial coefficient identities
Roots
Hypergeometric Equation
Symmetric Polynomials
Transcendental function
Single valued
Algebraic curve
Multivalued
Corollary
Trace
Differential equation
Ring
Generalise

Keywords

  • Algebraic curve
  • Algebraic function
  • Belyi cover
  • Binomial coefficient identity
  • Hypergeometric function
  • Imprimitive monodromy
  • Symmetric polynomial
  • Trinomial equation
  • Uniformization

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

The uniformization of certain algebraic hypergeometric functions. / Maier, Robert S.

In: Advances in Mathematics, Vol. 253, 01.03.2014, p. 86-138.

Research output: Contribution to journalArticle

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