论文标题
Apéry-Type系列和四级彩色Zeta值的一些变体
Some Variants of Apéry-Type Series and Level Four Colored Multiple Zeta Values
论文作者
论文摘要
在本文中，我们研究了涉及中央二项式系数的apéry-type系列\ begin {align*} \ sum_ {n_1> \ cdots> \ cdots> n_d> 0} \ frac1 {4^{n_1}}} \ frac {1} {n_1^{s_1} \ cdots n_d^{s_d}} \ end {align*}及其求和索引可能混合的平等和某些$> $''的变化被``$> $''替换为``$> $''''这些总和自然出现在Jegerlehner，Kalmykov和Veretin的工作中大量Feynman积分中。我们表明，所有这些总和可以表示为$ \ Mathbb Q $ - Q $ - 线性组合的颜色和/或虚构的彩色多个Zeta值的真实和/或假想部分，即在Unity的第四根根部的多个polygarithms的特殊值。我们还表明，相应的系列其中$ {\ binom {2n_1} {n_1}}}/4^{n_1} $被$ {\ binom {\ binom {\ binom {2n_1} {n_1} {n_1}}}^2/16^{n_1} $ for类似的$ 1/π以$ 1的$ 1/π表示。
In this paper, we study Apéry-type series involving the central binomial coefficients \begin{align*} \sum_{n_1>\cdots>n_d>0} \frac1{4^{n_1}}\binom{2n_1}{n_1} \frac{1}{n_1^{s_1}\cdots n_d^{s_d}} \end{align*} and its variations where the summation indices may have mixed parities and some or all ``$>$'' are replaced by ``$\ge$'', as long as the series are defined. These sums have naturally appeared in the calculation of massive Feynman integrals by the work of Jegerlehner, Kalmykov and Veretin. We show that all these sums can be expressed as $\mathbb Q$-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. We also show that the corresponding series where ${\binom{2n_1}{n_1}}/4^{n_1}$ is replaced by ${\binom{2n_1}{n_1}}^2/16^{n_1}$ can be expressed in a similar way except for a possible extra factor of $1/π$.
