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

By using various expansions of the digamma function and the method of residue computations, we study three variants of the linear Euler sums as well as related Hoffman’s double t-values and Kaneko-Tsumura’s double T-values and Kaneko-Tsumura’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. [ABSTRACT FROM AUTHOR]