Let us define $$F_n=\int f(z) |He_n(z)|^2 \, dz \, dz^*$$ is there any type of function $f$ could make that $F_n=0$ for $n\geq 2$ and $F_n>0$ for $n<2?$

$He_n(x)=2^{-\frac{n}{2}}H_n\left(\frac{x}{\sqrt{2}}\right)$ and $H_n$ is the usual physicist's Hermite polynomial.


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(Too long for the comment field.)

I understand that the question is not whether such functions exist (existence is easy to show) but how one can be produced explicitly.

It seems reasonable to take the ansatz $f(z)=g(|z|)$ where $g$ is a function on $[0,1]$ which is to be determined. The orthogonality relation is then recast in terms of real polynomials $p_n(r)=\int_0^{2\pi} |He_n(re^{it}|^2 r dt$. Precisely, $g$ must be orthogonal to $p_n$, $n>1$ on the positive halfline.Observe that the coefficients of $p_n$ are the squares of coefficients of $He_n$, only attached to different degrees (all odd). It is natural to suspect that $p_n$ or perhaps the polynomials $q_n$ defined by $p_n(x)=xq_n(x^2)$, are related to the Laguerre polynomials. If true, this would help with orthogonality but I could not see such a relation.


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