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

Let $p$ be an odd prime and $q=p^r$, $r\ge 1$. For positive integers $n$, let ${}_n{G}_n[\cdots {]}_q$ denote McCarthy’s $p$-adic hypergeometric functions. In this article, we prove an identity expressing a ${}_4{G}_4[\cdots {]}_q$ hypergeometric function as a sum of two ${}_2{G}_2[\cdots {]}_q$ hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transfomation that relates ${}_{n+2}{G}_{n+2}[\cdots {]}_q$ and ${}_n{G}_n[\cdots {]}_q$ hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ${}_4{G}_4[\cdots {]}_q$ and ${}_6{G}_6[\cdots {]}_q$ hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.