Ap\'ery-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I

TL;DR

This paper investigates Apéry-type series involving central binomial coefficients and their variants, demonstrating they can be expressed as rational linear combinations of colored multiple zeta values of level 4, thus linking special series to multiple polylogarithms at roots of unity.

Contribution

It establishes a connection between Apéry-type series with mixed parity indices and colored multiple zeta values of level 4, expanding understanding of their algebraic structure.

Findings

All studied series are rational linear combinations of colored multiple zeta values of level 4.

Series with squared binomial coefficients relate similarly to multiple zeta values.

Results unify various Apéry-type series within the framework of multiple polylogarithms at roots of unity.

Abstract

In this paper, we shall study A\'{e}ry-type series in which the central binomial coefficient appears as part of the summand. Let $b_n=4^n/\binom{2n}{n}$. Let $s_1,\dots,s_d$ be positive integers with $s_1\ge 2$. We consider the series \begin{align*} \sum_{n_1>\cdots>n_d>0} \frac{b_{n_1}}{n_1^{s_1}\cdots n_d^{s_d}} \end{align*} and the variants with some or all indices $n_j$ replaced by $2n_j\pm 1$ and some or all "$>$" replaced by "$\ge$", provided the series are defined. We can also replace $b_{n_1}$ by its square in the above series when $s_1\ge 3$. The main result is that all such series are $\mathbb{Q}$-linear combinations of the real and/or the imaginary parts of some colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity.