Closed-Form Evaluation of Two Ap\'ery-Like Series of Weight 4

TL;DR

This paper derives closed-form expressions for two new Apéry-like series of weight 4 involving harmonic numbers, providing insights into their mathematical structure and connections.

Contribution

It introduces and evaluates two novel Apéry-like series of weight 4 involving harmonic numbers of the form H_{2k}, expanding the understanding of such series.

Findings

Closed-form evaluations of two new Apéry-like series.

Connections established between the series and harmonic numbers.

Key results facilitate further research in series evaluation.

Abstract

This paper presents closed-form evaluations of two new Ap\'ery-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.