  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.
