Summation identities and transformations for hypergeometric series

Abstract: 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 $$Z_{\lambda}: x_1^d+x_2^d=d\lambda x_1x_2^{d-1}$$ over a finite field $\mathbb{F}p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z{\lambda}$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}p$ on the family $Z{\lambda}$ 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.