While inverting a Laplace transform using Post's inversion formula I found the following expression:
$$
\sum\_{k=1}^n S^n\_k \ x^k(\alpha)\_k
$$
where $S^n\_k$ is a Stirling number of second kind and $(\alpha)\_k$ is a Pochhammer symbol. This formula seems a mixture of the definition of these Stirling numbers and that of Touchard polynomials.


I tried without success to find an explicit expression for it. Does it exists? Or at least, is there an assymptotical expansion that for $n$ going to infinity?


Thank you!



Relations involving Stirling numbers of second kind
Let $f\_n$ denote your expression. Using the well-known formula $\sum\_n S\_k^n\frac{t^n}{n!}
=\frac{1}{k!}(e^t-1)^k$, we get the generating function
 $$ \sum\_{n\geq 0}f\_n\frac{t^n}{n!} = (1-x(e^t-1))^{-\alpha}. $$
This generating function suggests that there will be no simpler expression for $f\_n$ than its definition.


