Parametric Euler $T$-sums of odd harmonic numbers

TL;DR

This paper introduces parametric Euler T-sums, derives explicit formulas using complex analysis, and connects these to known multiple zeta values, advancing the understanding of harmonic number sums.

Contribution

It defines and analyzes parametric Euler T-sums, providing explicit formulas and linking them to Hoffman's and Kaneko-Tsumura's multiple zeta values.

Findings

Explicit formulas for linear parametric Euler T-sums

Connections to Hoffman's double t-values

Relations to Kaneko-Tsumura's double T-values

Abstract

In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the linear parametric Euler $T$-sums. Furthermore, by applying the results, we obtain explicit formulae for the Hoffman's (alternating) double $t$-values and Kaneko-Tsumura's (alternating) double $T$-values.