The polynomials are like:
$P_0 = 1\\ P_1 = -2+4x\\P_2 = 8-32x+16x^2\\P_3=-48 + 288 x - 288 x^2 + 64 x^3$
?
I have checked Legendre, Chebyshev, Hermite... Any help will be appreciated!
This comes from:
$\int_{-\infty}^{\infty} dy\; e^{iky} e^{-y^2/4} H_m(x+y/2)H_m(x-y/2)$
where, for polynomials, I did a replacement x=x^2+k^2.