In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive integers) I hadn't looked at the assumed approximations for the family of p-parametrized gamma-relatives (where p is nonnegative integer) $$ \begin{align} f_p(n) & =\exp \left(\sum_{k=0}^n \ln(1+k)^p \right) \\ & = 1^{\ln(1)^{p-1}}\cdot 2^{\ln(2)^{p-1}} \cdots n^{\ln(n)^{p-1}} \end{align}$$ where $p \gt 1$ .
I just looked at that treatize and would like to improve it with some knowlegde about the functions $f_p$ where $p \gt 1$ (for $p=1$ this is the factorial function).
Q: Has someone seen one of these functions being discussed elsewhere?
Here is some context: an older question at MO , an older question at MSE, the original text discussing this idea initially posted at the tetrationforum a very q&d or, a bit better written in "uncompleting the gamma", from page 13
