Appell’s hypergeometric series over finite fields

We define four functions [Formula: see text] and [Formula: see text] as finite field analogues of Appell series [Formula: see text] and [Formula: see text], respectively using purely Gauss sums in the spirit of finite field hypergeometric series introduced by McCarthy. We establish relations among [Formula: see text] and [Formula: see text] analogous to those satisfied by the classical Appell series. Recently, several people have defined finite field analogues of Appell series using integral representations of Appell series. We show that our functions [Formula: see text] and [Formula: see text] are closely related to those functions.