Some Variants of Apéry-Type Series and Level Four Colored Multiple Zeta Values
Ce Xu, Jianqiang Zhao
TL;DR
This work establishes that Apéry-type series built from central binomial coefficients, including squared versions, decompose into $\mathbb{Q}$-linear combinations of real and/or imaginary parts of colored multiple zeta values at level 4 (CMZV$^4$), with a potential extra $1/\pi$ factor for the squared case. The authors develop an extended Chen iterated-integral framework and a recursive tail-analysis to express these sums as iterated integrals (and then CMZVs) across multiple parity configurations, introducing block structures ($\alpha$, $\beta$, $\gamma$) to manage depth and convergence. They provide explicit results for many low-weight cases, prove weight-drop phenomena, and derive corollaries and examples illustrating CMZV reductions to constants like Catalan's constant $G$, zeta values, and polylogarithms at special points such as $(1+i)/2$. The methods illuminate connections between Apéry-type sums in quantum field theory and the arithmetic of level-4 CMZVs, with implications for broader classes of binomial-sum series and potential extensions to higher levels and alternating variants.
Abstract
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 $1/π$.