The Uniformization of Certain Algebraic Hypergeometric Functions

The hypergeometric functions nFn−1 are higher transcendental functions, but for certain parameter values they become algebraic. This occurs, e.g., if the defining hypergeometric differential equation has irreducible but imprimitive monodromy. It is shown that many algebraic nFn−1’s of this type can be represented as combinations of certain explicitly algebraic functions of a single variable, i.e., the roots of trinomial equations. This generalizes a result of Birkeland. Any tuple of roots of a trinomial equation traces out a projective algebraic curve, and it is determined when this curve is of genus zero, i.e., admits a rational parametrization. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic nFn−1’s. Even if the governing curve is of positive genus, it is shown how it may be possible to construct single-valued or multivalued parametrizations of individual algebraic nFn−1’s, by computation in rings of symmetric polynomials.