Summation identities and transformations for hypergeometric series

We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family Zλ:x1d+x2d=dλx1x2d-1over a finite field Fp. Salerno expresses the number of points over a finite field Fp on the family Zλ in terms of quotients of p-adic gamma functions under the condition that d| p- 1. In this paper, we first express the number of points over a finite field Fp on the family Zλ in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing d(d- 1) , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series. © 2017, Fondation Carl-Herz and Springer International Publishing AG.