I need to evaluate the following function and want to proceed analytically as far as possible:
$F(y) =e^{ i \beta \left ( y \frac{d}{d y} \right )^2} y \, e^{-y^2/2}$
My plan is to expand into power series in $\beta$ and indentify the polynomials
$\left (y \frac{d}{d y} \right )^k (y \, e^{-y^2/2})= p_k(y) \, y \, e^{-y^2/2}$
by the recursion relation I expect them to satisfy.
Is this a sound strategy? Is there a more direct way to identify polynomials $p_k(y)$ and compute their "generating function'' $F(y)$?