# Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p}$ been discussed anywhere?

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

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I haven't seen anything explicit about it before, but certainly it seems like the standard derivation of Stirling via the Euler-MacLaurin formula along with knowledge of the incomplete gamma function would give asymptotics. Is there something particular you're trying to find out? –  Steven Stadnicki Oct 4 '12 at 22:39
@Steven: I don't expect too much, it is just to do some completion of the discussion in my 3'rd link, which I just provided in my question. Well - perhaps there is something "nice" in it anyway... –  Gottfried Helms Oct 4 '12 at 22:58