4F3-Gaussian hypergeometric series and traces of Frobenius for elliptic curves

In this article, we obtain finite field analogues of classical summation identities connecting F-3-Appell series and F-4(3)-classical hypergeometric series. As an application, we establish a new summation formula satisfied by the F-4(3)-Gaussian hypergeometric series. We further express certain special values of F-4(3)-Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of F-4(3)-Gaussian hypergeometric series.