Fourier transforms of hermite polynomials

Stream: Is there code for X?

[Nicholas Wilson]

we have hermite polynomials under Mathlib.RingTheory.Polynomial.Hermite.{Basic,Gaussian} and we have Fourier transforms under Mathlib.Analysis.Fourier.FourierTransform. Do we have theorems about Fourier transforms of Hermite polynomials (namely that FT acts on "hermite n * gaussian" by multiplication byI^n)?