It's possible to generalize these series? $$\sum_{k=1}^{\infty}H_k^{(s)}x^k=\frac{\operatorname{Li}_s(x)}{1-x}$$ $$\sum_{k=1}^{\infty}H_k^2 x^k=\frac{\ln(1-x)^2+\operatorname{Li}_2(x)}{1-x}$$ Where: $$H_k:=\sum_{j=1}^{k}\frac{1}{j}\text{ are the harmonic numbers}$$ $$H_k^{(s)}:=\sum_{j=1}^{k}\frac{1}{j^s}\text{ are the generalized harmonic numbers}$$ $$\operatorname{Li}_s(z)\text{ is the polylogarithm}$$ I'd like to know if there are formulas for series like this: $$\sum_{k=1}^{\infty}\left(H_k^{(s)}\right)^n x^k\qquad \text{where }n\in\mathbb{N}$$
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1$\begingroup$ Second equality do you mean $H_k^{(2)}$ or $H_k^2$ ? $\endgroup$– EDXCommented May 6, 2023 at 12:11
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$\begingroup$ In the second equation it is $H_k^2$ $\endgroup$– Math AttackCommented May 11, 2023 at 10:18
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$\begingroup$ Have you tried an expression for $n=2,3,4$ at least. $\endgroup$– EDXCommented May 11, 2023 at 12:50
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