Evaluations of $ \sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series

TL;DR

This paper derives closed-form expressions for certain infinite series involving binomial coefficients and powers, expressing them in terms of cyclotomic multiple zeta values for specific parameter ranges.

Contribution

It introduces polylogarithmic reductions for classes of infinite sums and expresses these series using cyclotomic multiple zeta values of various levels, extending previous work.

Findings

Closed forms for series involving binomial coefficients and powers.

Representation of series in terms of cyclotomic multiple zeta values.

Applicable to series with parameters in (-27/4, 27/4).

Abstract

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$.