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

Ce Xu; Jianqiang Zhao

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arXiv · 2202.06195

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

Ce Xu·Jianqiang Zhao

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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.

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BibTeX RIS

Ce Xu, Jianqiang Zhao. 2022-02-13. Ap\'ery-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I. https://arxiv.org/abs/2202.06195

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