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

Let p be an odd prime and F-p be the finite field with p elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the p-adic setting which we denote by (3n-1)G(3n-1)(p,t), where n >= 1 and t is an element of F-p. These values are expressed in terms of numbers of zeros of certain polynomials over Fp. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zeros of these functions. For instance, if n is odd then we obtain zeros of (3n-1)G(3n-1)(p,t)=0 under certain condition on t. In contrast we show that if n is even then the function( 3n-1)G(3n-1)(p,t) has no zeros for any prime p apart from the trivial case when t=0.