A generalization of Bell numbers to arbitrary complex arguments

Question

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because $${\large\cup\,}\{\{1,2,3\}\}={\large\cup\,}\{\{1\},\{2,3\}\}={\large\cup\,}\{\{2\},\{1,3\}\}={\large\cup\,}\{\{3\},\{1,2\}\}={\large\cup\,}\{\{1\},\{2\},\{3\}\}.$$ Is there a nice generalization of Bell numbers to arbitary complex arguments, similar to how the number of permuations of $n$ items (factorial) is generalized to the gamma function $\Gamma(z)$?

Answer 1

As was mentioned in the comments, there are several possible generalizations:
1. Dobiński formula: $$B_x=\sum_{k=1}^\infty\frac{k^x}{e\,k!}$$ 2. Cesàro integral representation: $$B_x=\frac{2\,\Gamma(x+1)}{\pi\,e}\int_0^{\pi } \sin (t\,x)\sin \left(e^{\cos t}\sin\sin t\right)e^{e^{\cos t}\cos\sin t}\,dt$$

Interestingly, the results from these two formulae agree only for positive integer $x$, but are different for fractional $x$.

Answer 2

One can use Ramanujan's master theorem. As remarked in the comments, we have

$$B(x) := e^{e^x-1} = \sum_{n=0}^\infty B_n\frac{x^n}{n!}.$$

The function $F(x) = B(-x)-e^{-1}$ is of rapid decay as $x \to +\infty$, and therefore the integral

$$L(F,s):= \frac{1}{\Gamma(s)}\int_0^\infty F(x) x^s \frac{dx}{x}$$

converges for every $s>1$. By a standard argument, we can rewrite this as

$$L(F,s):= \frac{1}{\Gamma(s)(1-e^{2\pi i s})}\int_C F(x) x^s \frac{dx}{x}$$

where $C$ is the keyhole contour. The function $L(F,s)$ now makes sense for all complex $s$. At $s=-n$, where $n$ is a positive integer, the residue theorem shows that

$$L(F,-n) = \frac{(-1)^n}{n}\frac{d^n}{dx^n}\Big|_{x=0}F(x) = B_n/n.$$

Therefore the function $G(s)=sL(F, -s)$ satisfies $G(n)=B_n$ for every $n>0$.

It would be interesting to see whether this gives the same function as the one suggested by Lucian in the comments!