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Harmonic series with polygamma functions. (English) Zbl 1424.40044

Summary: The paper is about evaluating in closed form the following classes of series involving the product of the \(n\)th harmonic number and the polygamma functions \[ \begin{aligned} S_k &= \sum_{n=1}^\infty H_n\left(\zeta(k)-1-\frac{1}{s^k}-\dots -\frac {1}{n^k}\right)=\frac{(-1)^k}{(k-1)!}\sum_{n=1}^\infty H_n\varPsi^{(k-1)}(n+1),\quad k\geqslant 3,\\T_k &= \sum_{n=1}^\infty nH_n\left(\zeta(k)-1-\frac{1}{s^k}-\dots -\frac {1}{n^k}\right)=\frac{(-1)^k}{(k-1)!}\sum_{n=1}^\infty nH_n\varPsi^{(k-1)}(n+1),\quad k\geqslant 4,\\ \text{and}\\R_k &= \sum_{n=1}^\infty H_n^2\left(\zeta(k)-1-\frac{1}{s^k}-\dots -\frac {1}{n^k}\right)=\frac{(-1)^k}{(k-1)!}\sum_{n=1}^\infty H_n^2\varPsi^{(k-1)}(n+1),\quad k\geqslant 3,\end{aligned} \] where \(k\) is an integer.

MSC:

40G10 Abel, Borel and power series methods
40A05 Convergence and divergence of series and sequences
33B15 Gamma, beta and polygamma functions
11M35 Hurwitz and Lerch zeta functions
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References:

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