On alternating analogues of Tornheim's double series. II
Let \[ R(p,q,r)=\sum_{m,n=1}^\infty\frac{(-1)^n}{m^pn^q(m+n)^r}, \] \[ \mathfrak{T}_{b_1,b_2}(p,q,r)=\sum_{m,n=0}^\infty 1/{(2m+b_1)^p(2n+b_2)^q(2m+2n+b_1+b_2)^r}, \] where \(b_1,b_2\in\{1,2\}\). The author proves: Theorem. Suppose \(k,l,d\in\mathbb{N}\) with \(d\geq 2\) and \(k+l+d\equiv 1 \pmod 2\). Then \(R(k,l,d)\) and \(\mathfrak{T}_{b_1,b_2}(k,l,d)\) can be expressed as polynomials in \(\{\zeta(j)\mid 2\leq j\leq k+l+d\}\) with rational coefficients, where \(\zeta(s)\) is Riemann's zeta function. Part I, see Proc. Am. Math. Soc. 131, No. 12, 3633--3641 (2003; Zbl 1045.11057).
scientific article; zbMATH DE number 5636931