On variants of the Euler sums and symmetric extensions of the Kaneko-Tsumura conjecture

TL;DR

This paper investigates variants of Euler sums and extends the Kaneko-Tsumura conjecture using residue methods and expansions of the digamma function, revealing new symmetric identities involving special constants.

Contribution

It introduces new variants of Euler sums and establishes symmetric extensions of the Kaneko-Tsumura conjecture with detailed evaluations and identities involving Bernoulli and Genocchi numbers.

Findings

Established several symmetric extensions of the Kaneko-Tsumura conjecture.

Derived convolution identities involving Bernoulli and Genocchi numbers.

Provided explicit evaluations of special cases with mathematical constants.

Abstract

By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and establish several symmetric extensions of the Kaneko-Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and they are verified by the derivative polynomials of hyperbolic tangent.