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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!

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Maybe you want to use generating functions here? The sum in question should be representable as a n-th coefficient in some generating function. For $S^n_k$ some generating functions are known (mathworld.wolfram.com/StirlingNumberoftheSecondKind.html, especially (14)-(16)). Pochhammers can arise from some operation on GF's (see Wilf's Generatingfunctionology) –  Leonid Petrov Apr 8 '13 at 23:37

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up vote 4 down vote accepted

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.

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I was thinking about falling Pochhammer symbols so that changes the answer to $(1+x(e^t-1))^\alpha$, but that helps already. Thank you! –  guaraqe Apr 9 '13 at 6:41

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