Apéry-Type Series with Summation Indices of Mixed Parities and Colored Multiple Zeta Values, I

2022/02/13 by Ce Xu, Jian‐Qiang Zhao, Xu, Ce +1
Mathematics · #11B37 #11B65 #11M32 #33B30 #44A05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.06195

openalex publication_date 2022/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let b_n=4ⁿ/\binom2nn. Let s₁,…,s_d be positive integers with s₁≥ 2. We consider the series ∑_{n1gt;\cdotsgt;ndgt;0} \fracb_n1n₁^s₁⋯ n_d^s_d and the variants with some or all indices n_j replaced by 2n_j± 1 and some or all ">" replaced by "≥", provided the series are defined. We can also replace b_n1 by its square in the above series when s₁≥ 3. The main result is that all such series are ℚ-linear combinations of the real and/or the imaginary parts of some colored multiple zeta values of level 4, i.e., multiple polylogarithms evaluated at 4th roots of unity.

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