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

Cetin Hakimoglu-Brown

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 $Γ$-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.

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

Abstract

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

Paper Structure

This paper contains 7 sections, 1 theorem, 34 equations.

Table of Contents

  1. Introduction
  2. A motivating result
  3. Background
  4. Joyce and Zucker's special values of hypergeometric series
  5. Main results
  6. A new $_2F_1$-evaluation
  7. Conclusion

Key Result

Theorem 1

Gosper's formula in mainGosper holds for all values $b$ such that $b$ such that the left-hand side of mainGosper converges and such that its lower parameter is not a an integer in $\mathbb{Z}_{\leq 0}$ (cf. Vidunas2002). $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (2)

  • Theorem 1
  • proof