Summation identities and transformations for hypergeometric series -- II

Abstract

McCarthy defined a function in terms of quotients of the $p$-adic gamma functions which can best be described as an analogue of hypergeometric series in the $p$-adic setting. We find six transformations and three summation identities for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums. We also find a transformation and two summation identities for the Greene's finite field hypergeometric series. Finally, we obtain some special values of the Greene's finite field hypergeometric series as an application of our main results.