Abstract
Let p be an odd prime and [Formula: see text] be the finite field with p elements. This paper focuses on the study of the values of a generic family of hypergeometric functions in the p-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 p-adic analogues of classical hypergeometric identities. Namely, we obtain p-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 n is odd, we characterize t for which [Formula: see text] has zeros. In contrast, we show that if n is even, then the function [Formula: see text] has no zeros for any prime p apart from the trivial case when [Formula: see text].
Published Version
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