On the Evaluation of Apéry-Like Series Involving Multiple $t$-Harmonic Star Sums

We evaluate, by elementary means, a new family of Apéry-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.

💡 Research Summary

The paper investigates a new family of Apéry‑like series that involve multiple t‑harmonic star sums of even weight. For a non‑negative integer j the authors consider the series
\