Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series F2* and F4* in terms of 4F3-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series F2 and F4. We also deduce an expression for the finite field Appell series F4* in terms of 3F2-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series F4*. (c) 2023 Elsevier Inc. All rights reserved.