Alternating Apéry-Type Series and Colored Multiple Zeta Values of Level Eight

Abstract

Apéry-type (inverse) binomial series have appeared prominently in the calculations of Feynman integrals in recent years. In their previous work, the authors showed that a few large classes of the non-alternating Apéry-type (inverse) central binomial series can be evaluated using colored multiple zeta values of level four (i.e., special values of multiple polylogarithms at the fourth roots of unity) by expressing them in terms of iterated integrals. In this sequel, the authors will prove that for several classes of the alternating versions they need to raise the level to eight. Their main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.

Keywords

Apéry-type series / Colored multiple zeta values / Binomial coefficients / Iterated integrals / 11M32 / 11B65 / 11B37 / 44A05 / 11M06 / 33B30

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