I'm looking for an "easy" proof of the asymptotic expansion of Hermite functions ($f_n(x)=H_n(x)e^{-x^2/2}$ where $H_n$ is the Hermite polynomials).
The asymptotic expansion is $$ f_n(x) \sim_{n \rightarrow \infty} \sqrt{2}(\frac{2n}{e})^{n/2}cos(\sqrt{2n}x-n\frac{\pi}{2})$$
I don't find any proof in "orthogonal polynommials" of Szego. In "orthogonal functions" of Sansone, I find this: $$f_{2n}(x)=H_{2n}(0)cos(\sqrt{4n+1}x)+h(2n,x)$$
where $H_{2n}(0)cos(\sqrt{4n+1}x) \sim_{n \rightarrow \infty} \sqrt{2}(\frac{4n}{e})^{n}cos(\sqrt{4n}x-n\frac{\pi}{2})$
And $h(2n,x)=-\frac{1}{\sqrt{4n+1}}\int_0^x t^2f_{2n}(t)sin(\sqrt{4n+1}(t-x))dt$
With $\mid{\int_0^x t^2f_{2n}(t)sin(\sqrt{4n+1}(t-x))dt} \mid \leq \mid{x}\mid^{5/2} ||f_{2n}||_{L^2}$
But it isn't sufficient to conclude. Can anyone help me ?
Thanks