It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of polygamma functions by writing $$H_n:=H_{n}^{(1)}=\int_0^1\frac{1-x^n}{1-x}dx=\gamma+\psi_{0}(n+1)$$ and for integer $r\ge 2$ $$H_{n}^{(r)}=\zeta(r) -(-1)^r\frac{\psi_{r-1}(n+1)}{\Gamma(r)}.$$ I suppose that, upon adding some mild condition (in a similar way as for the Bohr Mollerup theorem), this extension is unique.
Now what I would like to find is a (the) natural extension of $$G_n^{(r)}:=\sum_{k=1}^n\frac1{k}H_k^{(r)}.$$
Motivation: I have identified a family $\{f(n)\}$ of continued fractions which (numerically) seem to have for $n\in\mathbb N$ the closed form $G_n^{(2)}$ as their value. Now, for those continued fractions, $n$ may also be a non integer, and at least e.g. for $n=\dfrac12$, there "should" (morally...) be a closed form.
So stated equivalently: for the $r=2$ case, it would be about finding a "direct" form in terms of $n$ for $$\sum_{k=1}^n\frac{\psi_{1}(k+1)}{k}=-\sum_{k=1}^n\int_0^1\frac{x^k}k\frac{ \log x}{1-x}dx=\int_0^1 \frac{1-x^n}{1-x} \mathrm{Li}_2(1-x)~dx.$$ Straightforward GP/Pari code for $G_n^{(2)}$:
for(n=1,10, s=sum(k=1,n,sum(j=1,k,1/j^2)/k); print(n," ",s*1.," = ",s))
1 1.00000000000000 = 1
2 1.62500000000000 = 13/8
3 2.07870370370370 = 449/216
4 2.43460648148148 = 4207/1728
5 2.72732870370370 = 589103/216000
6 2.97589351851852 = 642793/216000
7 3.19186452596912 = 236478859/74088000
8 3.38279228248839 = 2004994517/592704000
9 3.55387758595134 = 56872731439/16003008000
10 3.70885435906800 = 59352825979/16003008000
