Some Variants of Apéry-Type Series and Level Four Colored Multiple Zeta Values

Abstract: In this paper, we study Ap\'{e}ry-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 $1/\pi$.