Elementary proofs of Zagier's formula for multiple zeta values and its odd variant

Li, Lai; Lupu, Cezar; Orr, Derek

doi:10.48550/arxiv.2201.09262

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First Variant With Odd Summation Indices1

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2022

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“…In [10], T. Murakami first proved the analog of Zagier's 2-3-2 formula of MZVs for multiple t-values t(2 a , 3, 2 b ). Thereafter, several other proofs have appeared in the literature, see for example [8].…”

Section: First Variant With Odd Summation Indicesmentioning

confidence: 99%

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

Xu¹,

Zhao²

2022

Preprint

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In this paper, we shall study Aéry-type series in which the central binomial coefficient appears as part of the summand. Let b n = 4 n / 2n n . Let s 1 , . . . , s d be positive integers with s 1 ≥ 2. We consider the seriesand 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 n 1 by its square in the above series when s 1 ≥ 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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Section: First Variant With Odd Summation Indicesmentioning

confidence: 99%