5
$\begingroup$

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

  • 0
    @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...2012-10-04

0 Answers 0