Harmonic sums and rational multiples of zeta functions (Q1758799)

In this paper the author develops some identities associated with harmonic numbers and gives the integral representations such as \[ \begin{multlined} \sum_{n=1}^\infty \frac{t^n \sum_{r=1}^{an} \frac{1}{r+j}}{n^3{an+j \choose j} {bn+k \choose k} {cn+l \choose l}} \\ = -abct\int_0^1 \int_0^1 \int_0^1 \frac{(1-x)^j(1-y)^k(1-z)^l x^{a-1}y^{b-1}z^{c-1} \log(1-x)}{1-tx^ay^bz^c} dx\,dy\,dz \end{multlined} \] where \(a, b, c\) are real positive numbers, \(j,k,l \geq 0\) and \(|t| < 1\). In particular, \[ \sum_{n=1}^\infty \frac{H_n}{n^3 {n+k \choose k}^2} = -\int_0^1 \int_0^1 \int_0^1 \frac{[(1-y)(1-z)]^k \ln(1-x)}{1-xyz} dx\,dy\,dz. \]