Explicit formulas of Euler sums via multiple zeta values. (November 2020)

• Record Type: Journal Article
• Authors: Xu, Ce; Wang, Weiping
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 109
• Page End: 127
• Publication Date: 2020-11
• Subjects: 40A25 -- 11B83 -- 11M32; Compositions -- Euler sums -- Harmonic numbers -- Multiple zeta values -- Permutations; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.06.009
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Flajolet and Salvy pointed out that every Euler sum is a Q -linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein–Borwein–Girgensohn theorems, and the Flajolet–Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of S r m, q, S r m, q ¯, S r ¯ m, q and S r ¯ m, q ¯, are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight w ≤ 11 for non-alternating case and of weight w ≤ 6 for alternating case.