An Euler non linear sum

Let $\mathbb{N} \ni m \geq 2$. Then the following formula

$$\sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(m)}}{n^m}=\frac{\zeta^2(m)+\zeta(2m)}{2}$$

holds.

Solution

Hidden message

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We are not stating a proof here, but only presenting the result. However if someone insists on how to do this, then he might begin with the function $$f(z)=\frac{\psi_1^m(-z)}{z^{m-1}}$$