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

Let p be an odd prime and q = pr,r >= 1. For positive integers n,let n Gn[<middle dot><middle dot><middle dot>]qdenoteMcCarthy's p-adic hypergeometric function. In this paper, we prove an identity express-ing a4G4[<middle dot><middle dot><middle dot>]qhypergeometric function as a sum of two2G2[<middle dot><middle dot><middle dot>]qhypergeometric func-tions. This identity generalizes some known identities satisfied by the finite field hyper-geometric functions. We also prove a transformation that relatesn+2Gn+2[<middle dot><middle dot><middle dot>]qandnGn[<middle dot><middle dot><middle dot>]qhypergeometric functions. Next, we express the trace of Frobenius of ellipticcurves in terms of special values of4G4[<middle dot><middle dot><middle dot>]qand6G6[<middle dot><middle dot><middle dot>]qhypergeometric functions.Our results extend the recent works of Tripathi and Meher on the finite field hypergeo-metric functions to wider classes of primes