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I have noticed an interesting generating function involving Harmonic Numbers.

$$\sum_{n=1}^{\infty}H_nx^n=\frac{\ln(1-x)}{x-1}$$

But, I have not seen a generating function involving second-order Harmonic numbers, such as

(1) $\sum_{n=1}^{\infty}H_n^{(2)}x^n$

(2) $\sum_{n=1}^{\infty}(H_n^{(2)})^2x^n$

(3) $\sum_{n=1}^{\infty}(H_n^{(2)})^kx^n$

I was wondering if these generating functions are known, and how can I go about finding them? Thanks.

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You can find a formula on Wikipedia: $\sum \limits _{n=1} ^\infty H_n ^{(m)} x^n = \frac {\mathrm{Li}_m (z)} {1-z}$, where $\mathrm{Li}_m$ is the polylogarithm. I doubt that you will be able to find something involving more elementary functions.

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  • $\begingroup$ I think that beyond what you answered, there will not be any closed form expression. $\endgroup$ – Claude Leibovici Jul 26 '15 at 8:29

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