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

2022/05/02 by Ce Xu, Jian‐Qiang Zhao, Xu, Ce +1
Mathematics · #11B37 #11B65 #11M06 #11M32 #33B30 #44A05 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2205.01000

openalex publication_date 2022/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Apéry-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 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 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.

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