= Solution
By part a the transformed function solves the same equation with $\mu_n$, and by part c its <polynomial> factor has degree $n$. The uniqueness proved in part b therefore gives $q_n=c_np_n$. Comparing their leading <coefficients> gives
$$
\boxed{\widehat{f_n}=c(-i)^nf_n.}
$$
For the unitary convention used above, the Gaussian integral gives $c=1$, so \b[$c_n=(-i)^n$]. With the unnormalized phase-$e^{-i\lambda x}$ convention, $c=\sqrt{2\pi}$ instead. This proves that <Hermite functions are Fourier eigenfunctions>, with the normalization-dependent scalar fully specified.
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