p-Adic hypergeometric functions and the trace of Frobenius of elliptic curves

Let p be an odd prime and Formula: see text, Formula: see text. For positive integers n, let Formula: see text denote McCarthy’s p-adic hypergeometric function. In this paper, we prove an identity expressing a Formula: see text hypergeometric function as a sum of two Formula: see text hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transformation that relates Formula: see text and Formula: see text hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of Formula: see text and Formula: see text hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.