## The Uniformization of Certain Algebraic Hypergeometric Functions

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Maier, Robert S.

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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.

Comment: 58 pages, accepted by Advances in Mathematics

Comment: 58 pages, accepted by Advances in Mathematics

##### Keywords

Mathematics - Commutative Algebra, Mathematics - Algebraic Geometry, Mathematics - Classical Analysis and ODEs, 33C20 (Primary), 33C80, 14H20, 14H45, 05E05 (Secondary)