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

TL;DR

This paper extends a contiguity relation method to derive closed-form values of Appell's hypergeometric series $F_1$, leading to new algebraic values of Gauss's $_2F_1$ with no free parameters.

Contribution

It applies a contiguity relation approach to Appell's series $F_1$ to find new closed-form expressions and algebraic values of $_2F_1$, expanding the understanding of hypergeometric series.

Findings

Derived several closed-form $F_1$ series

Obtained $_2F_1$ values with no free parameters

Provided new algebraic $_2F_1$ examples

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 $_2F_1$ 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 $_2F_1$ with no free parameters. Some of these results provide new examples of algebraic values of $_2F_1$. Key Words and Phrases: Gauss's hypergeometric series, algebraic value, Appell's hypergeometric series, hypergeometric identity.