On a ${}_2{F}_1\left(\frac{1}{4}\right)$-identity due to Gosper

Abstract

It is only in exceptional cases that a ${}_2{F}_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z\in \{\pm 1,\frac{1}{2}\}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $\Gamma$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for ${}_2{F}_1$-series of the desired form. We apply this approach using a ${}_2{F}_1\left(\frac{1}{4}\right)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_2{F}_1$-series of convergence rate $(\frac{172872}{185039}{)}^2$. With regard to extant research on so-called ``strange'' ${}_2{F}_1$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument.

11 pages