Parametric Euler sum identities (Q819709)

The paper deals with parametrized classes of multiple sums studied in relation to double Euler sums and Euler's reflection formula. The main result is the identity \[ \sum_{n=1}^\infty{1 \over {n(n-x)}}\sum_{m=1}^{n-1}{1 \over {m-x}} = \sum_{n=1}^\infty{1 \over {n^2(n-x)}} \] which is valid for all \(x \in\mathbb C\setminus Z^+\). A more complex identity is established for the double sum \[ \sum_{n=1}^\infty\bigg({n \over {n^2 + y^2}}+\sum_{m=1}^{n-1}{2m \over {m^2+y^2}}\bigg){1 \over {n^2+x^2}} \]