# Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k$

Is there a generating function for

$$\tag{1}\sum_{k\geq 1} H^{(k)}_n x^ k$$

I know that

$$\tag{2}\sum_{k\geq 1} H^{(k)}_n x^n= \frac{\operatorname{Li}_k(x)}{1-x}$$

But notice in (1) the fixed $n$.

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Let $\psi(x)=\frac{\Gamma'}{\Gamma}(x)=\frac{d}{dx}\log\Gamma(x)$ be the digamma function. For $N$ a positive integer, we have $$\psi(x+N)-\psi(x)=\sum_{j=0}^{N-1}\frac{1}{x+j}$$ (this follows from $x\Gamma(x)=\Gamma(x+1)$ and induction).