Interpolate the number of arrangements in a set

Question

I am working the integral $$\int_0^\infty e^{x(k-\alpha) - e^x} dx$$ where $k$ is a positive integer and $\alpha$ a positive real.

WolframAlpha shows that for $\alpha=0$ and $k=1,\ldots,7$ the integral has a value proportional to $1,2,16,65,326,1957,13700$ which I identified in OEIS as the sequence $a_n$ of the number of arrangements of a set of size $n$, that is $$ \sum_{k=0}^n \frac{n!}{k!} $$

Now, I want to put $\alpha > 0$, and the question is if there are an evident way to interpolate this function in the same fashion that the factorial is by the Gamma function.

A direct integration works for me also :)

Thanks in advance!

Answer 1

Mathematica v9.0 gives $\int_0^{\infty } \exp (k x-\exp (x)) \, dx$ = $\Gamma (k,1)$ with
$\Gamma (k,1)$ = $\frac{\sum _{i=0}^{k-1} \frac{(k-1)!}{i!}}{e}$ on condition the real part of k >0
Obviously, increasing your $\alpha$ is equivalent to decreasing 'my' k in the result above. Remark that $\Gamma (z,1)$ is defined for complex z, not just integer x. See http://mathworld.wolfram.com/IncompleteGammaFunction.html