How Can the Hermite Polynomial Identity Be Proven?

[appelberry]

Does anyone know how to prove the following identity:

$$\Sigma_{k=0}^{n}\left(\stackrel{n}{k}\right) H_{k}(x)H_{n-k}(y)=2^{n/2}H_{n}(2^{-1/2}(x+y))$$

where $$H_{i}(z)$$represents the Hermite polynomial?

 

[JSuarez]

I would try induction on n.