  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.
