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

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.