Values of p-adic hypergeometric functions and p-adic analogue of Kummer’s linear identity

Abstract

Let [Formula: see text] be an odd prime and [Formula: see text] be the finite field with [Formula: see text] elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the [Formula: see text]-adic setting, which we denote by [Formula: see text] where [Formula: see text] and [Formula: see text]. These values are expressed in terms of numbers of zeros of certain polynomials over [Formula: see text]. These results lead to certain [Formula: see text]-adic analogues of classical hypergeometric identities. Namely, we obtain [Formula: see text]-adic analogues of particular cases of a Gauss’s theorem and a Kummer’s theorem. Moreover, we examine the zeros of these functions. For example, if [Formula: see text] is odd, we characterize [Formula: see text] for which [Formula: see text] has zeros. In contrast, we show that if [Formula: see text] is even, then the function [Formula: see text] has no zeros for any prime [Formula: see text] apart from the trivial case when [Formula: see text].