Special Values of the Hypergeometric Series

In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series F ( a , b ; c ; x ) F(a,b;c;x) ; we show that values of F ( a , b ; c ; x ) F(a,b;c;x) at some points x x can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of F ( a , b ; c ; x ) F(a,b;c;x) that can be obtained with this method. We find that this set includes almost all previously known values and many previously unknown values.