I'm trying to find the following limit:
$$ \underset{n \to \infty}{\textrm{lim}} \frac{\sqrt[3]{n}}{2^n n! \sqrt{\pi}}\ \int_{\sqrt{2n+1}}^{\infty}\textrm{H}^2_n(x) e^{-x^2}dx $$
Where H is a hermite polynomial (the physicist kind). I'm pretty sure the limit is
$$ \frac{1}{3^{2/3}\Gamma^2(1/3)} $$
This is based on an asymptotic formula using Airy functions.
Any thoughts on how to find the limit?
If it is any help, the limit can also be represented as
$$ \underset{n\rightarrow \infty}{\textrm{lim}}\frac{\sqrt[3]{n}}{2 \cdot n!}\frac{d^n }{dx^n} \left .\frac{\textrm{erfc}\left ( \sqrt{\tfrac{1-x}{1+x}} \sqrt{2n+1}\right )}{1-x} \right |_{x=0} $$