3
votes
1answer
246 views

The relationship between Stirling number of the second kind and the polylogarithm

It is shown here on Mathworld's page on Stirling number of the second kind that $$ \sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z}) $$ where $S(n,k)$ is Stirling number of the ...
1
vote
1answer
562 views

Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral $\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ ...