Four families of summation formulas involving generalized harmonic numbers (Q1696805)

The generalized harmonic number means \(H_n^{<\ell>}(x)= \sum_{k=1}^n \frac{1}{(x+k)^{\ell}}.\) For a natural number \(s=0,1,2\), the paper produces closed form expressions for \[ \sum_{k=1}^n k^sH_k^2(x),\quad \sum_{k=1}^n k^sH_k^3(x),\quad \sum_{k=1}^n k^sH_k(x) H_k^{<2>}(x), \] and \[ \sum_{k=1}^n k^sH_k^2(x) H_k^{<2>}(x), \] using Abel's transformation, but the authors would be able to do such calculations for larger \(s\) well. These results generalize a number of published special cases.