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Journal of the Ramanujan Mathematical Society

Volume 36, Issue 2, June 2021  pp. 93–107.

Gamma functions and sum formulas of multiple zeta values

Authors:  Kwang-Wu Chen and Minking Eie
Author institution:Department of Mathematics, University of Taipei, No.~1, Ai-Guo West Road, Taipei 10048, Taiwan

Summary:  For a pair of positive integers n, k with ⌈ n/3 ⌉ ≤ k ≤ ⌊ n/2 ⌋, let E{2,3} (n,k) be the sum of multiple zeta values of weight n and depth k with arguments 2 or 3, i.e., E{2,3} (n,k) = ∑ {|{α}| = n} {α{j} = 2 or 3} ζ(α{1}, α{2}, …, α{k}). Hoffman conjectured that the set {ζ(α{1}, … , α{r}) : r ∈ N, α{i} ∈ {2,3}} is a Q-basis for the vector space spanned by all MZVs, and Brown has proved that Hoffman's conjectured basis is in fact a spanning set. Therefore it is desirable to investigate sum formulas such as E{{2,3}} (n, r). In this paper, we evaluate the general form E{s, t} (n,k) in terms of ζ(2), ζ(3), … , ζ(n) with integers s, t rather than 2, 3. The proofs are based on some relations with Gamma functions. Moreover we get a lot of similar sum formulas of multiple zeta values and multiple zeta-star values as applications.

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