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.