[Paper Review] Two Variants of Euler Sums

Ce Xu, Weiping Wang|arXiv (Cornell University)|Jun 15, 2019

Advanced Mathematical Identities22 references4 citations

TL;DR

This paper introduces and evaluates two variants of Euler sums—Euler $T$-sums and alternating Euler $T$-sums—using contour integration and the residue theorem. It derives explicit formulas for these sums in terms of Hoffman $t$-values and multiple zeta values, establishes a duality formula for Kaneko-Tsumura's conjecture, and provides closed forms for triple $t$-values and even-weighted Kaneko-Tsumura $T$-zeta values.

ABSTRACT

For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators. Like the Euler sums, the Euler $T$-sums can be evaluated according to the Contour integral and residue theorem. Using this fact, we obtain explicit formulas for Euler $T$-sums with repeated arguments analogous to those known for Euler sums. Euler $T$-sums can be written as rational linear combinations of the Hoffman $t$-values. Using known results for Hoffman $t$-values, we obtain some examples of Euler $T$-sums in terms of (alternating) multiple zeta values. Moreover, we prove an explicit formula of triple $t$-values in terms of zeta values, double zeta values and double $t$-values. We also define alternating Euler $T$-sums and prove some results about them by the Contour integral and residue theorem. Furthermore, we define another Euler type $T$-sums and find many interesting results. In particular, we give an explicit formulas of triple Kaneko-Tsumura $T$-values of even weight in terms of single and the double $T$-values. Finally, we prove a duality formula of Kaneko-Tsumura's conjecture.

Motivation & Objective

• To define and systematically study Euler $T$-sums, a variant of classical Euler sums restricted to odd denominators.
• To extend the theory to alternating Euler $T$-sums and other Euler-type $T$-sums, including Kaneko-Tsumura $T$-zeta values.
• To establish explicit evaluations of these sums using contour integration and residue calculus, linking them to Hoffman $t$-values and multiple zeta values.
• To prove a duality formula for Kaneko-Tsumura's conjecture, connecting various $T$-sums and $t$-values.
• To provide closed-form expressions for triple $t$-values and even-weighted Kaneko-Tsumura $T$-zeta values in terms of simpler zeta and $t$-values.

Proposed method

• Define Euler $T$-sums $T_{p_1ar{p}_k,q}$ as sums over odd denominators, with harmonic-like numerators $h_{n-1}^{(p)}$.
• Apply contour integration and the residue theorem to evaluate $T$-sums, leveraging the properties of the digamma function and its derivatives.
• Use the relation between $T$-sums and Hoffman $t$-values: every Euler $T$-sum is a rational linear combination of $t$-values of the same weight and depth at most $k+1$.
• Derive explicit formulas for triple $t$-values in terms of zeta values, double zeta values, and double $t$-values.
• Introduce alternating Euler $T$-sums and evaluate them using the same contour integration framework.
• Prove a duality formula for Kaneko-Tsumura’s conjecture by combining functional equations and residue identities involving Hurwitz zeta functions and $\Psi^{(m-1)}$ functions.

Experimental results

Research questions

• RQ1 How can Euler $T$-sums with repeated arguments be evaluated explicitly, analogous to known results for classical Euler sums?
• RQ2 What is the relationship between Euler $T$-sums and Hoffman $t$-values, and can $T$-sums be expressed as rational linear combinations of $t$-values?
• RQ3 Can a duality formula be established for Kaneko-Tsumura’s conjecture involving $T$-sums and $t$-values?
• RQ4 What are the closed-form evaluations of triple $t$-values and even-weighted Kaneko-Tsumura $T$-zeta values in terms of simpler zeta values?
• RQ5 How do alternating Euler $T$-sums behave, and can they be evaluated using the same analytic techniques as non-alternating ones?

Key findings

• The paper derives an explicit formula for triple $t$-values in terms of zeta values, double zeta values, and double $t$-values.
• It proves that Euler $T$-sums of weight $w$ and degree $k$ are rational linear combinations of Hoffman $t$-values of weight $w$ and depth at most $k+1$.
• For $p=2, q=3$, the identity $3T(2,4) + 2T(3,3) = \frac{\pi^6}{64} - \frac{49}{8}\zeta^2(3)$ is established as a concrete example.
• A duality formula is proven for Kaneko-Tsumura’s conjecture, relating sums of $T(m+i,q+j)$ and $T(p+i,q+j)$ to $\widetilde{t}(p+q+m-1)$ and products of zeta and $t$-values.
• The relation $\widetilde{S}_{p,q} = 2^{p+q-2}T(q,p)$ and $T_{p,q} = 2^{p+q}t(q,p)$ is used to connect double Kaneko-Tsumura $T$-values with double Hoffman $t$-values.
• Corollary 5.4 gives a specific evaluation: $\widetilde{S}_{2,2} + 2\widetilde{S}_{1,3} - T_{2,2} = \frac{\pi^4}{12}$, illustrating the method’s effectiveness.