On the Evaluation of Ap\'ery-Like Series Involving Multiple $t$-Harmonic Star Sums

TL;DR

This paper evaluates a new family of Apéry-like series involving multiple t-harmonic star sums of even weight, providing explicit closed-form formulas using elementary methods and special functions.

Contribution

It introduces a novel approach to evaluate Apéry-like series with multiple t-harmonic star sums, deriving explicit formulas as finite sums of Dirichlet beta values.

Findings

Explicit closed-form evaluations of the series.

Derivation of several concrete examples.

Use of elementary methods for complex series evaluation.

Abstract

We evaluate, by elementary means, a new family of Ap\'ery-like series involving multiple $t$-harmonic star sums of even weight. Using trigonometric expansions, inverse tangent integrals, and binomial recurrences, we obtain explicit closed-form evaluations of these series as finite alternating sums of products of Dirichlet beta values. Several explicit examples are derived as corollaries.