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