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I have been searching for identities involving generalized harmonic numbers \begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*} I found several identities in terms of $H_n^{(1)}$, but I am looking for some interesting identities for $H_n^{(2)}$. Does anyone know of any identities know of any nontrivial identities for $H_n^{(2)}$? I found some listed on Wikipedia, but this list is not comprehensive. Thanks for your help.

  1. integral identities
  2. summation identities
  3. recursive identities
  4. in terms of another function
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I discovered a Generalized Harmonic Summation Identity some time ago.

$$ \sum_{r=1}^{n} \dfrac{H_{r} ^{(m)}}{r^m} = \dfrac{1}{2} \left( [H_{n}^{(m)}]^2 + H_{n}^{(2m)} \right) \quad ; \quad m \geq 1 $$

I have posted a proof here.

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There is a nice list and a set of references at mathworld. Additionally, I discovered this one while writing a thesis on the Riemann Zeta function.

$$\sum_{n=1}^\infty \frac{H_n^{(s)}} {n^s}=\frac{\zeta(s)^2+\zeta(2s)}{2}.$$

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