English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we study Ap\'{e}ry-type series involving the central binomialcoefficients \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 mixedparities and some or all "$>$" are replaced by "$\ge$", as long as the seriesare defined. We show that all these sums can be expressed as $\mathbb Q$-linearcombinations of the real and/or imaginary parts of the colored multiple zetavalues at level four, i.e., special values of multiple polylogarithms at fourthroots 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$1/\pi$.

Ap\'{e}ry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, II