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!