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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'' $F(y)$?

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    Why do you have $\left( y \frac{d}{y} \right)^2$ in the exponent? Wouldn't that just cancel out to $d^2$?2012-03-04
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    Sorry, it's a typo. I meant the derivative with respect to y.2012-03-04
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    Considering $F$ as a function of both $y$ and $\beta$, we obtain the PDE $$\begin{cases} \left(\frac{d}{d\beta}-i\left(y\frac{d}{dy}\right)^2\right)F=0 \\ \\ \left. F\, \right|_{\beta=0}=ye^{-y^2/2}. \end{cases}$$ It's possible this could help.2012-03-04
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    @anon: With the substitution $y=e^{-x/2}$ it leads to the time-dependent Schrodinger equation of free 1d motion with a specific initial condition. Solving the latter using Green's function results in an integral which is the origin of my question. I keep a hope that progress can be made in explicit evaluation of $F$.2012-03-04
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    I'm implenting the strategy, looks like these are Sloane A039755: https://oeis.org/A039755 , will post an answer if this leads to a solution.2012-03-04

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