Special values of Gauss's hypergeometric series derived from Appell's\n series $F_1$ with closed forms

In a previous work ([Eb]), the author proposed a method employing contiguity\nrelations to derive hypergeometric series in closed form. In [Eb], this method\nwas used to derive Gauss's hypergeometric series $_2F_1$ possessing closed\nforms. Here, we consider the application of this method to Appell's\nhypergeometetric series $F_1$ and derive several $F_1$ possessing closed forms.\nMoreover, analyzing these $F_1$, we obtain values of $_2F_1$ with no free\nparameters. Some of these results provide new examples of algebraic values of\n$_2F_1$.\n Key Words and Phrases: Gauss's hypergeometric series, algebraic value,\nAppell's hypergeometric series, hypergeometric identity.\n