论文标题
Apéry-type系列具有混合奇偶和有色多个Zeta值的求和索引，i
Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I
论文作者
论文摘要
在本文中，我们将研究Aéry-type系列，其中中央二项式系数显示为汇总的一部分。令$ b_n = 4^n/\ binom {2n} {n} $。令$ s_1，\ dots，s_d $为正整数，$ s_1 \ ge 2 $。我们考虑系列\ begin {align*} \ sum_ {n_1> \ cdots> n_d> 0} \ frac {b_ {n_1}} {n_1^{n_1^{s_1} \ cdots n_d n_d^{s_d}}}}}}}}}}} and n_ n_ n _ n_j n_j {align {align {Aliant = $ 2N_J \ pm 1 $和一些或全部“ $> $”代替了“ $ \ ge $”，只要定义了该系列。当$ s_1 \ ge 3 $时，我们还可以通过上述系列中的平方替换$ b_ {n_1} $。主要结果是所有此类系列都是$ \ Mathbb {q} $ - 真实和/或某些彩色的多个ZETA值4级的ZETA值的线性组合，即，在Unity的第四根基础上评估的多个Polyrogarithms。
In this paper, we shall study Aéry-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.
