The power series needs the be simplified is given as \begin{equation} \sum_{p=k}^n\frac{\mu^p}{p!}\binom{p}{k}(-1)^{p-k}. \end{equation} Can it be simplified in a more compact form?
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Why have you edited in the way you did, did my hint help? My small effort seems redundant now you have removed any reference to Touchard and Stirling numbers, indeed, it may be determined you have asked a different question. – Sep 13 '17 at 13:18
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@Kevin Sorry, I posted a different question by mistake and correct it by editing it. The other problem is https://math.stackexchange.com/questions/2425043/express-a-power-series-in-a-closed-form?noredirect=1#comment5009269_2425043. Sorry for the annoying. – seuhoww Sep 14 '17 at 01:31
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The Touchard polynomials can be written in terms of the Stirling numbers of the second kind, viz, $$T_n(x) = \sum_{k=0}^n S_{2}(k;n)x^k$$
Where $$S_2(k;n) = \frac{1}{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}j^k$$
Implying that $$T_n(x) = \sum_{k=0}^n \frac{x^k}{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}j^k$$