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Two closed-form evaluations for the generalized hypergeometric function ${}_4F_3(\frac1{16})$

Arjun K. Rathie, Mykola A. Shpot

TL;DR

This work derives two closed-form evaluations for ${}_4F_3(1/16)$ by exploiting an even/odd decomposition of ${}_3F_2$ and leveraging known ${}_3F_2(±1/4)$ results. The resulting formulas express ${}_4F_3(1/16)$ in terms of $\psi'(1/3)$ and $\pi^2$, and enable a faster-converging infinite-sum representation for $pi^2$ when combined. The approach reveals connections to Ramanujan-type sums and the Clausen function, enriching the interplay between hypergeometric sums, binomial/harmonic identities, and special functions. These findings extend explicit evaluation techniques for ${}_pF_q$ functions and may have applications in mathematical physics and analytic number theory.

Abstract

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $π^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds.

Two closed-form evaluations for the generalized hypergeometric function ${}_4F_3(\frac1{16})$

TL;DR

This work derives two closed-form evaluations for by exploiting an even/odd decomposition of and leveraging known results. The resulting formulas express in terms of and , and enable a faster-converging infinite-sum representation for when combined. The approach reveals connections to Ramanujan-type sums and the Clausen function, enriching the interplay between hypergeometric sums, binomial/harmonic identities, and special functions. These findings extend explicit evaluation techniques for functions and may have applications in mathematical physics and analytic number theory.

Abstract

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function of the argument . This is achieved by means of separating a generalized hypergeometric function into even and odd components, together with the use of two known results for available in the literature. As an application, we obtain an interesting infinite-sum representation for the number . Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds.
Paper Structure (5 sections, 1 theorem, 31 equations)

This paper contains 5 sections, 1 theorem, 31 equations.

Table of Contents

  1. Introduction
  2. Even and odd components of ${}_pF_q(z)$
  3. Two closed-form evaluations for ${}_4F_3(\frac{1}{16})$
  4. Relation to Ramanujan's notebooks, other functions, binomial and harmonic sums
  5. Concluding remarks

Key Result

Theorem 1

The following two closed-form evaluations for the function ${}_4F_3(\frac{1}{16})$ hold true: and where $\psi(z)$ is the psi-function, the logarithmic derivative of the Euler gamma function $\Gamma(z)$.

Theorems & Definitions (2)

  • Theorem 1
  • proof