$_(,_)$ is a martingale

Question

Please explain how to prove that $_(,_)$ is a martingale.

H_n(x,y)=∂^n/∂α^n exp(αx−1/2(α^2)y) at α=0.

Xαt=exp(αWt−1/2(α^2)t)=∑n=0∞α^n/n! Hn(Wt,t).

I found a lot of literature just claiming this, but nobody proves it anywhere.