Tornheim type series and nonlinear Euler sums

Evaluations of Euler-Type Sums of Weight $$\le 5$$≤5

“…In 1998, Flajolet and Salvy [12] used the contour integral representations and residue computation to show that the quadratic sums S p 1 p 2 ,q are reducible to linear sums and zeta values when the weight p 1 + p 2 + q is even and p 1 , p 2 > 1. The best results to date are due to Xu and Wang et al, see the most recent papers [17,20,22]. In [20,22], we proved that all Euler sums of weight ≤ 8 are reducible to Q-linear combinations of single zeta monomials with the addition of {S 2,6 } for weight 8.…”

“…The best results to date are due to Xu and Wang et al, see the most recent papers [17,20,22]. In [20,22], we proved that all Euler sums of weight ≤ 8 are reducible to Q-linear combinations of single zeta monomials with the addition of {S 2,6 } for weight 8. For weight 9, all Euler sums of the form S s 1 ···s k ,q with q ∈ {4, 5, 6, 7} are expressible polynomially in terms of zeta values.…”

“…When x → 1, then the sum S p 1 p 2 ···pm,p (x) reduces to the classical Euler sum, which is defined by ([12,20,22]) S p 1 p 2 ···pm,p :=…”

Explicit formulas of Euler sums via multiple zeta values

“…Recently, rapid progress has been made in this field. Using the Bell polynomials, generating functions, integrals of special functions, multiple zeta (star) values, the Stirling sums and the Tornheim type series, we study the (alternating) Euler sums systematically [42][43][44][46][47][48][49][50][51]53,54]. As a consequence, the evaluation of all the unknown Euler sums up to the weight 11 are presented, and a basis of Euler sums of weight 3 ≤ w ≤ 11 is…”

Some results on q-harmonic number sums

“…The relationship between the values of the Riemann zeta function and the classical Euler sums W 0 (m; p) (or S m;p ) has been studied by many authors (for example, see [7][8][9][10][11][12][13][14][15][16] and the references therein). So far, surprisingly little work has been done on q-analogues of Euler sums.…”