Factorial-like function that behaves like $ 1\times \sqrt{2} \times \sqrt[3]{3} \times \dots \times \sqrt[n-1]{n-1} \times \sqrt[n]{n} $? The Barnes G-function is a generalization of the factorial funcion which grows a lot faster.  The factorial:
$$ n! = 1 \times 2 \times 3 \times \dots \times (n-1) \times  n  $$
Then it is possible to write a super-factorial:
$$ n!! = 1! \times 2! \times 3! \times \dots \times (n-1)! \times  n!
= 1^n \times 2^{n-1} \times \dots \times (n-1)^2 \times n^1  $$
For a given combinatorics function $f(n)$ we can try to extend to $f(x)$ with $n < x < n+1$.  The super-factorial can be continued to the Barnes G-function. 
How can I get a function that behaves like the product of successive $n$-th roots?
$$ f(n) =  1\times \sqrt{2} \times \sqrt[3]{3} \times \dots \times \sqrt[n-1]{n-1} \times \sqrt[n]{n} $$
I tried looking amongst the multiple gamma functions but there is no negative multiple Gamma functions... now what?
 A: One may first observe that
$$
\begin{align}
f(n)&=1\times \sqrt{2} \times \sqrt[3]{3} \times \dots \times \sqrt[n-1]{n-1} \times \sqrt[n]{n}
\\&=\prod_{k=1}^n k^{\large \frac1k}
\\&=e^{ \sum_{k=1}^n\!\frac{\ln k}k}.
\end{align}
$$ We are then led to consider $\displaystyle \sum_{k=1}^n \frac{\ln k}k$. 
One may recall the Stieltjes constants and the generalized Stieltjes constants,
$$
\begin{align}
\gamma_1=&\lim_{N \to \infty}\left(\sum_{k=1}^N \frac{\ln k}k- \int_{1}^N \frac{\ln x}x\:dx\right)
\\\gamma_1(a)=&\lim_{N \to \infty}\left(\sum_{k=0}^N \frac{\ln (k+a)}{k+a}-\int_{0}^N \frac{\ln (x+a)}{x+a}\:dx\right),\quad  a>0,
\end{align}
$$  then one may observe that, for $n\ge1$,
$$
\sum_{k=1}^n \frac{\ln k}k= \lim_{N \to \infty}\left(\sum_{k=1}^N \frac{\ln k}k- \frac{\ln^2 N}2\right)-\lim_{N \to \infty}\left(\sum_{k=1}^N \frac{\ln (k+n)}{k+n}- \frac{\ln^2 (N+n)}2\right)
$$ giving

$$
1\times \sqrt{2} \times \sqrt[3]{3} \times \dots \times \sqrt[n-1]{n-1} \times \sqrt[n]{n}=e^{\Large\gamma_1-\gamma_1(n+1)}, \qquad n\ge1.
$$

Some results on the Stieltjes constants : Coffey (2005), Adell (2010), Fekih-Ahmed (2014), Paris (2015).
