Alternating Ap\'{e}ry-Type Series and Colored Multiple Zeta Values of Level Eight

TL;DR

This paper extends the evaluation of Apéry-type series related to Feynman integrals by demonstrating that alternating series can be expressed using colored multiple zeta values of level eight, employing hyperbolic trigonometric forms.

Contribution

It introduces a method to evaluate alternating Apéry-type series using level eight colored multiple zeta values with hyperbolic forms, expanding previous non-alternating results.

Findings

Alternating series are expressible via level eight colored multiple zeta values.

Hyperbolic trigonometric forms replace ordinary ones in the evaluation process.

The approach generalizes previous non-alternating series evaluations.

Abstract

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