On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper

TL;DR

This paper introduces a new integration-based method for evaluating special values of $_2F_1$ hypergeometric series with rational parameters, applying it to a Gosper identity to compute a series with a notably large rational argument.

Contribution

The paper presents a novel approach for constructing special $_2F_1$-series values and applies it to evaluate a series associated with Gosper's identity, expanding understanding of hypergeometric evaluations.

Findings

Evaluated a $_2F_1$-series with a large rational argument (172872/185039)^2.

Developed an integration-based method for constructing special $_2F_1$-series values.

Identified a $_2F_1$-evaluation with the largest numerator/denominator in its argument among known 'strange' evaluations.

Abstract

It is only in exceptional cases that a $_2F_1(z)$-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 $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. 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.