Mixed Parity Variants of Apéry-Type Binomial Series and Level Four Colored Multiple Zeta Values

In this paper, we study an Apéry-type series involving the central binomial coefficients ${\sum }_{n_1>\dots >n_d>0}{\displaystyle \frac{1}{4^{n_1}}}\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)n_1^{-s_1}\dots n_d^{-s_d}$ and its variations where the summation indices may have mixed parities and some or all “>” are replaced by “≥”, 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. For example, our main theorem shows that when $n_1^{-s_1}$ is replaced by ${\left(2n_1\right)}^{-s_1}$ and other $n_j^{-s_j}$’s are replaced by either ${\left(2n_j\right)}^{-s_j}$ or ${(2n_j+1)}^{-s_j}$, then all the colored multiple zeta values can be chosen to have the same weight $s_1+\dots +s_d$, but the weights of these values are only bounded by $s_1+\dots +s_d$ for general variant Apéry-type series of mixed parities. We also show that the corresponding series where $\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)/4^{n_1}$ is replaced by ${\left({\displaystyle \genfrac{}{}{0pt}{}{2n_1}{n_1}}\right)}^2/{16}^{n_1}$ can be expressed in a similar way except for a possible extra factor of $1/\pi$, with the weight of the colored multiple zeta values similarly bounded.