### 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 language | English (US) |
---|---|

Pages (from-to) | 86-138 |

Number of pages | 53 |

Journal | Advances in Mathematics |

Volume | 253 |

DOIs | |

State | Published - Mar 1 2014 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 253, pp. 86-138. https://doi.org/10.1016/j.aim.2013.11.013

}

TY - JOUR

T1 - The uniformization of certain algebraic hypergeometric functions

AU - Maier, Robert S

PY - 2014/3/1

Y1 - 2014/3/1

N2 - 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.

AB - 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.

KW - Algebraic curve

KW - Algebraic function

KW - Belyi cover

KW - Binomial coefficient identity

KW - Hypergeometric function

KW - Imprimitive monodromy

KW - Symmetric polynomial

KW - Trinomial equation

KW - Uniformization

UR - http://www.scopus.com/inward/record.url?scp=84890854622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890854622&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2013.11.013

DO - 10.1016/j.aim.2013.11.013

M3 - Article

AN - SCOPUS:84890854622

VL - 253

SP - 86

EP - 138

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -