Special Values of the Hypergeometric Series

In this paper, the author presents 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, the author gets identities for the hypergeometric series $F(a,b;c;x)$ and shows that values of $F(a,b;c;x)$ at some points $x$ can be expressed in terms of gamma functions, together with certain elementary functions. The author tabulates the values of $F(a,b;c;x)$ that can be obtained with this method and finds that this set includes almost all previously known values and many previously unknown values.