Let $ P_k(x) $ be a homogeneous harmonic polynomial with order $ k\geq 1 $ on $ \mathbb{R}^n $. Show that $$ \mathcal{F}(P_k(x)e^{-\pi|x|^2})(\xi)=i^{-k}P_k(\xi)e^{-\pi |\xi|^2}. $$ Here $ \mathcal{F}(f)(\xi)=\int_{\mathbb{R}^n}f(x)e^{-2\pi ix\cdot\xi}dx $.
Here is my try. By letting $ f(\xi)=\mathcal{F}(P_k(x)e^{-\pi|x|^2})(\xi) $, I can show that $$ f(\xi)(4\pi^2|\xi|^2+2\pi n)+\sum_{j=1}^{n}2\pi\xi_j\partial_jf(\xi)+\Delta f(\xi)=0. $$ If $ g(\xi)=f(\xi)e^{\pi |\xi|^2} $, it can be got that $ \Delta g(\xi)=0 $. Also, it can be observed that $ g(\xi) $ is a polynomial with order $ k $. However, I do not know how to go on, can you give me some hints or references?