[Paper Review] Some evaluation of cubic Euler sums

Ce Xu|arXiv (Cornell University)|May 17, 2017

Advanced Mathematical Identities8 references3 citations

TL;DR

This paper develops a novel approach based on Tornheim-type series to evaluate cubic Euler sums, proving that sums like $ S_{1^2m,m} $ and $ S_{1(2l+1)^2,2l+1} $ reduce to zeta values, quadratic, and linear sums. It further shows that certain depth-four multiple zeta values reduce to depth-three ones, and establishes reducibility of alternating cubic sums to alternating quadratic and linear sums.

ABSTRACT

P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Moreover, we prove that the two combined sums involving multiple zeta values of depth four \[\sum\limits_{\left\{ {i,j} ight\} \in \left\{ {1,2} ight\},i e j} {ζ\left( {{m_i},{m_j},1,1} ight)}\quad { m and}\quad \sum\limits_{\left\{ {i,j,k} ight\} \in \left\{ {1,2,3} ight\},i e j e k} {ζ\left( {{m_i},{m_j},{m_k},1} ight)} \] can be expressed in terms of multiple zeta values of depth $\leq 3$, here $2\leq m_1,m_2,m_3\in \N$. Finally, we evaluate the alternating cubic Euler sums ${S_{{{\bar 1}^3},2r + 1}}$ and show that it are reducible to alternating quadratic and linear Euler sums. The approach is based on Tornheim type series computations.

Motivation & Objective

• To develop a systematic method for evaluating cubic Euler sums $ S_{1^2m,p} $ and $ S_{1l_1l_2,l_3} $ using Tornheim-type series.
• To determine when cubic Euler sums are reducible to zeta values, quadratic, and linear Euler sums.
• To investigate the reducibility of multiple zeta values of depth four to those of depth ≤3.
• To analyze the structure of alternating cubic Euler sums and prove their reducibility to alternating quadratic and linear sums.
• To extend the Flajolet-Salvy theorem on Euler sum reducibility to cubic and alternating cases.

Proposed method

• Utilizes Tornheim-type series computations as the core analytical framework for evaluating cubic Euler sums.
• Applies contour integral representations and residue calculus to relate sums to multiple zeta values.
• Employs harmonic number identities and polylogarithm function expansions to decompose complex sums.
• Uses known evaluations of linear and quadratic sums as building blocks for higher-order reductions.
• Applies algebraic manipulation to express depth-four multiple zeta values in terms of lower-depth ones.
• Leverages known results on alternating Euler sums and multiple zeta values to derive closed forms for alternating cubic sums.

Experimental results

Research questions

• RQ1 Can the cubic Euler sums $ S_{1^2m,m} $ and $ S_{1(2l+1)^2,2l+1} $ be expressed in terms of zeta values, quadratic, and linear sums?
• RQ2 Are the combined sums $ extstyleigsum_{i eq j}inom{ ext{m}_i, ext{m}_j,1,1}{ ext{m}_i, ext{m}_j,1,1} $ and $ extstyleigsum_{i eq j eq k}inom{ ext{m}_i, ext{m}_j, ext{m}_k,1}{ ext{m}_i, ext{m}_j, ext{m}_k,1} $ reducible to multiple zeta values of depth ≤3?
• RQ3 Can the alternating cubic Euler sum $ S_{ar{1}^3,2r+1} $ be reduced to alternating quadratic and linear Euler sums?
• RQ4 What is the structure of depth-four multiple zeta values, and can they be simplified to depth-three ones?
• RQ5 How do Tornheim-type series facilitate the evaluation of nonlinear Euler sums beyond the Flajolet-Salvy framework?

Key findings

• The cubic sums $ S_{1^2m,m} $ and $ S_{1(2l+1)^2,2l+1} $ are reducible to zeta values, quadratic, and linear Euler sums.
• The combined sum $ extstyleigsum_{i eq j}inom{ ext{m}_i, ext{m}_j,1,1}{ ext{m}_i, ext{m}_j,1,1} $ with $ m_i eq m_j $, $ m_i,m_j eq 1 $, reduces to multiple zeta values of depth ≤3.
• The sum $ extstyleigsum_{i eq j eq k}inom{ ext{m}_i, ext{m}_j, ext{m}_k,1}{ ext{m}_i, ext{m}_j, ext{m}_k,1} $ with $ m_i,m_j,m_k eq 1 $ also reduces to depth-three multiple zeta values.
• The alternating cubic Euler sum $ S_{ar{1}^3,2r+1} $ is reducible to alternating quadratic and linear Euler sums.
• Explicit evaluation of $ S_{ar{1}^3,3} $ yields a closed form involving $ ext{Li}_5(1/2) $, powers of $ \ln 2 $, zeta values, and $ \zeta^*(\bar{5},\bar{1}) $, $ \zeta^*(4,\bar{2}) $.
• The identity $ \zeta^*(\bar{4},2) + 4\zeta^*(\bar{5},1) = -\frac{1105}{192}\zeta(6) + \frac{3}{4}\zeta^2(3) $ is confirmed via sum decomposition and known relations.