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

Abstract

In a previous work ([Eb]), the author proposed a method employing contiguity relations to derive hypergeometric series in closed form. In [Eb], this method was used to derive Gauss's hypergeometric series ${}_2{F}_1$ possessing closed forms. Here, we consider the application of this method to Appell's hypergeometetric series $F_1$ and derive several $F_1$ possessing closed forms. Moreover, analyzing these $F_1$, we obtain values of ${}_2{F}_1$ with no free parameters. Some of these results provide new examples of algebraic values of ${}_2{F}_1$. Key Words and Phrases: Gauss's hypergeometric series, algebraic value, Appell's hypergeometric series, hypergeometric identity.

arXiv comment16 pages, 5 tables, 1 figure