I have been playing around a lot with Euler sums lately, and in an attempt to derive results not explicitly mentioned in the papers I've read, I wanted to get the OGF of $$\sum_{n = 1}^{\infty} H_n^4 z^n$$ and stumbled upon these three questions all outlining different ways of how you could go about trying to evaluate this, i.e.
- Representing $H_{n+1}^k$ in terms of $H_n^k$, then integrate and index shift.
- Using the integral representation of $H_n$, then expressing higher powers by iterated integrals and swapping integral and sum.
- Evaluating a recursively defined integral sequence.
I have tried calculating the easier example $\sum_{n = 1}^{\infty} H_n^2 z^n$ by hand through all of these ways, but except for the first method I haven't succeeded in showing the identity given in the first link. The method given there is the most straightforward, but not easily generalizable, since naive evaluation of the sums $\sum_{n = 1}^{\infty} \frac{H_n^p}{n^q} z^n$ by integration introduces some constant terms that explode when splitting apart the integral. Mathematica also has trouble getting recognizable closed forms following these approaches. So:
Is there a general way of solving or reducing this problem to some smaller problem (that could subsequently be treated with a CAS)? Am I being way too optimistic?
or as a start
Can we extract a closed form for $\sum_{n = 1}^{\infty} H_n^5 z^n$?
I can also paste the closed forms of the OGF for $H_n,H_n^2,H_n^3$ and $H_n^4$ to this question if requested.
