Closed-Form Evaluation of Two Apéry-Like Series of Weight 4

TL;DR

This work gives closed-form evaluations for two Apéry-like weight-4 series involving harmonic numbers $H_{2k}^{(2)}$ and $H_{2k}^2$ with central-binomial weighting and the factor $4^k$. By establishing a suite of lemmas that connect harmonic-number sums to integrals, generating functions, and polylogarithms, the author expresses the sums in terms of Catalan's constant $G$, $\zeta(4)$, $\operatorname{Li}_4(1/2)$, and logarithmic terms. The approach links integral representations, polylogarithmic constants, and telescoping relations to obtain explicit closed forms for the two weight-4 series. These results extend the catalog of closed-form Apéry-like sums and illustrate techniques for evaluating similar harmonic-number series with central-binomial denominators.

Abstract

This paper presents closed-form evaluations of two new Apéry-like series of weight $4$ that involve harmonic numbers of the form $H_{2k}$. Several key results are derived and subsequently used to establish connections to the main series.

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• proof