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

Abstract

Let p be an odd prime and p 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 3n-1G3n-1(p,t), where n ? 1 and t p. These values are expressed in terms of numbers of zeros of certain polynomials over p. 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 3n-1G3n-1(p,t) has zeros. In contrast, we show that if n is even, then the function 3n-1G3n-1(p,t) has no zeros for any prime p apart from the trivial case when t = 0. � 2024 World Scientific Publishing Company.