[Paper Review] Evaluations of Euler type sums of weight $\leq$ 5

Ce Xu|arXiv (Cornell University)|Apr 9, 2017

Advanced Mathematical Identities18 references4 citations

TL;DR

This paper evaluates Euler-type sums of the form $ S_{p_1 p_2 \cdots p_m, p}(x) = \sum_{n=1}^\infty \frac{H_n^{(p_1)} H_n^{(p_2)} \cdots H_n^{(p_m)}}{n^p} x^n $ at $ x = \frac{1}{2} $ for total weight $ \leq 5 $. Using linear relations between sums and known special functions, it proves that all such sums can be expressed as rational linear combinations of products of zeta values, polylogarithms, and $ \log(2) $, providing explicit closed forms for weight-5 cases and identities for weight-6 sums.

ABSTRACT

Let $p,p_1,\ldots,p_m$ be positive integers with $p_1\leq p_2\leq\cdots\leq p_m$ and $x\in [-1,1)$, define the so-called Euler type sums ${S_{{p_1}{p_2} \cdots {p_m},p}}\left( x ight)$, which are the infinite sums whose general term is a product of harmonic numbers of index $n$, a power of $n^{-1}$ and variable $x^n$, by \[S_{p_1 p_2 \cdots p_m, p}(x) := \sum_{n = 1}^\infty \frac{H_n^{(p_1)} H_n^{(p_2)} \cdots H_n^{(p_m)}} {n^p} x^n \quad (m\in \mathbb{N} := \{1,2,3,\ldots\}), \] where $H_n^{(p)}$ is defined by the generalized harmonic number. Extending earlier work about classical Euler sums, we prove that whenever $p+p_1+\cdots+p_m \leq 5$, then all sums ${S_{{p_1}{p_2} \cdots {p_m},p}}\left( 1/2 ight)$ can be expressed as a rational linear combination of products of zeta values, polylogarithms and $\log(2)$. The proof involves finding and solving linear equations which relate the different types of sums to each other.

Motivation & Objective

• To determine when generalized Euler sums of weight ≤5 at $ x = \frac{1}{2} $ can be expressed in terms of elementary constants.
• To extend classical Euler sum evaluations to weighted sums involving generalized harmonic numbers and $ x^n $.
• To establish a systematic method for expressing these sums as rational linear combinations of zeta values, polylogarithms, and $ \log(2) $.
• To derive explicit closed forms for all $ S_{p_1 p_2 \cdots p_m, p}\left(\frac{1}{2}\right) $ with total weight $ \leq 5 $.
• To explore identities among higher-weight sums (weight 6) using the same method.

Proposed method

• The method relies on constructing and solving systems of linear equations relating different types of Euler-type sums.
• It uses known integral representations and generating functions involving logarithms and polylogarithms to derive identities.
• The approach leverages the connection between multiple harmonic sums and multiple zeta values (MZVs), including alternating variants.
• The paper employs the generating function identity $ \sum_{n=1}^\infty H_n^{(p)} \frac{x^n}{n^q} = \sum_{k=1}^\infty \frac{1}{k^p} \sum_{n=1}^\infty \frac{x^n}{n^q (n+k)} $ to derive relations.
• It uses the known identity $ S_{2n-1,1}(z) = \mathrm{Li}_{2n}(z) + \frac{1}{2} \sum_{k=1}^{2n-1} (-1)^{k+1} \mathrm{Li}_k(z) \mathrm{Li}_{2n-k}(z) $ to derive closed forms for $ S_{5,1}(1/2) $.
• The evaluation of weight-6 sums is based on deriving and solving linear relations among sums of the same weight, such as $ \mathrm{Li}_3^2(1/2) = 2S_{3,3}(1/2) + \cdots - 20\mathrm{Li}_6(1/2) $.

Experimental results

Research questions

• RQ1 Can all Euler-type sums of weight ≤5 at $ x = \frac{1}{2} $ be expressed as rational linear combinations of zeta values, polylogarithms, and $ \log(2) $?
• RQ2 What linear relations exist between Euler-type sums of the same weight, particularly at $ x = \frac{1}{2} $?
• RQ3 How can higher-weight sums (e.g., weight 6) be related to lower-weight sums and special functions?
• RQ4 What closed-form expression can be derived for $ S_{5,1}(1/2) $?
• RQ5 Are there systematic methods to evaluate sums involving products of generalized harmonic numbers and $ x^n $ for $ x = \frac{1}{2} $?

Key findings

• All Euler-type sums of weight ≤5 at $ x = \frac{1}{2} $ are expressible as rational linear combinations of products of zeta values, polylogarithms, and $ \log(2) $.
• The sum $ S_{5,1}\left(\frac{1}{2}\right) $ has a closed form involving $ \mathrm{Li}_6(1/2) $, $ \mathrm{Li}_5(1/2)\log(2) $, $ \mathrm{Li}_4(1/2)\zeta(2) $, $ \mathrm{Li}_4(1/2)\log^2(2) $, $ \zeta^2(3) $, and powers of $ \log(2) $.
• The identity $ \mathrm{Li}_3^2(1/2) = 2S_{3,3}(1/2) + 6S_{2,4}(1/2) + 12S_{1,5}(1/2) - 20\mathrm{Li}_6(1/2) $ holds and enables evaluation of $ S_{3,3}(1/2) $.
• The sum $ \zeta(4,1,1;1/2) $ is expressed as a combination of $ S_{1^2,4}(1/2) $, $ S_{2,4}(1/2) $, $ S_{1,5}(1/2) $, and $ \mathrm{Li}_6(1/2) $, yielding a rational expression in $ \zeta(6) $, $ \zeta^2(3) $, $ \mathrm{Li}_5(1/2)\log(2) $, and powers of $ \log(2) $.
• The evaluation of $ S_{5,1}(1/2) $ is derived from identities (6.103)–(6.105), resulting in a closed form with 10 terms including $ \mathrm{Li}_6(1/2) $, $ \mathrm{Li}_5(1/2)\log(2) $, $ \zeta^2(3) $, and $ \log^6(2) $.
• The method successfully derives identities for weight-6 sums, such as $ \mathrm{Li}_2(1/2)\mathrm{Li}_4(1/2) = S_{4,2}(1/2) + 2S_{3,3}(1/2) + 4S_{2,4}(1/2) + 8S_{1,5}(1/2) - 15\mathrm{Li}_6(1/2) $.