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

TL;DR

This work addresses the evaluation of Apéry-like series built from finite multiple $t$-harmonic star sums of even weight, introducing an elementary method based on trig expansions, inverse tangent integrals, and binomial recurrences. The main result provides a closed form: for each $j\ge 0$, $\displaystyle \sum_{n=1}^{\infty} \frac{4^n}{n^2 \binom{2n}{n}} t_n^{\star}(\{2\}_j) = 8 \sum_{k=0}^{2j} (-1)^k \beta(k+1) \beta(2j-k+1)$, with explicit corollaries for small $j$ in terms of Dirichlet beta values and Catalan’s constant $G$. The paper further extends the method to harmonic-star sums $\zeta_n^{\star}(\{2\}_j)$, obtaining a parallel identity involving $\eta(2j+1)$ and presenting connections to known constants and recent conjectures (Genčev–Rucki), now resolved by Xu. Overall, the results deliver exact, elementary evaluations for a new family of Apéry-like series and illuminate their deep ties to special values of Dirichlet beta and related constants.

Abstract

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.

Theorems & Definitions (9)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Corollary 1