It is well known that harmonic numbers in general cannot be expressed through elementary functions. I am interested in following sums: \begin{equation} {\mathcal S}_\theta(n):=\sum\limits_{p=0}^n \binom{n}{p} (-1)^p H_{\frac{p}{\theta}} \end{equation} where $\theta$ is a strictly positive integer.Now, using the integral representation of harmonic numbers I have shown that : \begin{eqnarray} {\mathcal S}_1(n) &=& - \frac{1}{n} \\ {\mathcal S}_2(n) &=& - \frac{2}{n} + \Psi\left(\frac{1}{2},1,n\right) \\ \end{eqnarray} Here $\Psi$ is the Lerch zeta function. As it seems the result consists of two parts the first one having a simple closed form and the other one being non-expressable through elementary functions. Now, of course the question is what is the result for arbitrary values of $\theta$.


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