[Paper Review] Explicit formulas of Euler sums via multiple zeta values

Ce Xu, Weiping Wang|arXiv (Cornell University)|May 17, 2018

Advanced Mathematical Identities46 references3 citations

TL;DR

This paper presents explicit formulas that express Euler sums and alternating Euler sums as rational linear combinations of multiple zeta values (MZVs) and alternating MZVs, respectively, using permutations and compositions. The key contribution is a systematic method to reduce all such sums—up to weight 11 for non-alternating and weight 6 for alternating cases—to MZVs, with a corresponding Maple package for computation.

ABSTRACT

Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-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 $S_{r^m,q}$, $S_{r^m,\bar{q}}$, $S_{\bar{r}^m,q}$ and $S_{\bar{r}^m,\bar{q}}$, are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight $w\leq 11$ for non-alternating case and of weight $w\leq 6$ for alternating case.

Motivation & Objective

• To resolve the long-standing open problem of finding a general explicit formula that expresses all Euler sums as rational linear combinations of multiple zeta values (MZVs).
• To extend this framework to alternating Euler sums, showing they are reducible to alternating MZVs.
• To provide a unified theoretical basis that recovers famous results such as Euler's theorem, the Borwein–Borwein–Girgensohn theorems, and Flajolet–Salvy’s theorems.
• To compute and tabulate explicit evaluations of specific Euler sums (e.g., $S_{r^m,q}$, $S_{ar{r}^m,q}$) up to weight 11 for non-alternating and weight 6 for alternating cases.
• To develop a Maple package for automated evaluation of Euler sums based on the derived explicit formulas.

Proposed method

• Utilize permutations and compositions of integers to decompose the structure of Euler sums and derive explicit algebraic identities.
• Apply the quasi-shuffle algebra of nested sums to model the algebraic relations among harmonic numbers and zeta values.
• Express the numerator of the summand in the Euler sum as a linear combination of MZV basis elements via combinatorial decomposition.
• Use symbolic computation techniques to systematically reduce products of harmonic numbers to combinations of MZVs through recursive decomposition.
• Implement the derived formulas in a custom Maple package to compute all Euler sums of weight $ w \leq 11 $ (non-alternating) and $ w \leq 6 $ (alternating).
• Leverage known results on MZVs and alternating MZVs to validate and calibrate the reduction process.

Experimental results

Research questions

• RQ1 Can all Euler sums be explicitly expressed as rational linear combinations of multiple zeta values using a general formula?
• RQ2 To what extent can alternating Euler sums be reduced to alternating multiple zeta values?
• RQ3 How can the structure of harmonic number products in Euler sums be systematically decomposed using permutations and compositions?
• RQ4 What is the complete set of basis elements needed to express all Euler sums of a given weight?
• RQ5 Can a computational framework be developed to automate the evaluation of Euler sums up to a specified weight?

Key findings

• All non-alternating Euler sums of weight $ w \leq 11 $ are expressible as rational linear combinations of MZVs, with a complete basis provided for weights 3 to 11.
• All alternating Euler sums of weight $ w \leq 6 $ are reducible to alternating MZVs, and explicit evaluations are tabulated for 18 specific sums in Table 1.
• The method recovers classical results such as Euler’s formula $ \zeta(2,1) = \zeta(3) $, the Borwein–Borwein–Girgensohn theorems on quadratic sums, and Flajolet–Salvy’s general reduction theorems.
• The paper provides explicit closed-form expressions for special families such as $ S_{r^m,q} $, $ S_{ar{r}^m,q} $, $ S_{r^m,\bar{q}} $, and $ S_{\bar{r}^m,\bar{q}} $, with concrete evaluations given.
• A Maple package is developed and validated to compute all Euler sums of weight $ w \leq 11 $ (non-alternating) and $ w \leq 6 $ (alternating), enabling automated evaluation.
• The evaluations of specific sums like $ S_{1^4,\bar{2}} $, $ S_{1^2 2,\bar{2}} $, and $ S_{2^2,\bar{2}} $ are explicitly computed and tabulated with rational coefficients.