Apéry numbers, Jacobi sums, and special values of generalized p-adic hypergeometric functions

Abstract

We present some summation formulae for some special values of ratios of generalized p-adic hypergeometric functions in terms of the p-adic gamma function. By well-known methods such formulae yield expressions for roots of congruence zeta-functions in terms of Jacobi sums in the nonsingular case. We also express various formal-group congruences, including those involving Apéry numbers, in terms of p-adic hypergeometric functions at singular and nonsingular points of the associated differential equation. These congruences yield p-adic analytic formulae for unit roots of certain Hecke polynomials.