In some physics related problem, I found out the curious identity
$$\sum\limits\_{n\_1+n\_2+n\_3=n}\frac{n!}{n\_1!\,n\_2!\,n\_3!}\,H\_{2n\_1}(x)\,H\_{2n\_2}(y)\,H\_{2n\_3}(z)=\frac{H\_{2n+1}(r)}{2r},$$
where $H\_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$ are Hermite polynomials and
$r=\sqrt{x^2+y^2+z^2}$. Is this identity known?



Is this Hermite polynomial identity known?
If we define the generating functions $F(x,t)=\sum\_{n=0}^{\infty}H\_{2n}(x)\frac{t^n}{n!}$ and $G(x,t)=\sum\_{n=0}^{\infty}H\_{2n+1}(x)\frac{t^n}{n!}$ then your identity is equivalent to
$$F(x,t)F(y,t)F(z,t)=\frac{G\left(\sqrt{x^2+y^2+z^2},t\right)}{2\sqrt{x^2+y^2+z^2}}.$$
This is in turn an immediate corollary to the fact that we have
$$F(x,t)=\frac{1}{(1+4t)^{1/2}}\exp\left(\frac{4tx^2}{1+4t}\right)$$
$$G(x,t)=\frac{2x}{(1+4t)^{3/2}}\exp\left(\frac{4tx^2}{1+4t}\right)$$
and these have appeared in the literature in various contexts. [Here](http://iopscience.iop.org/article/10.1088/1674-1056/23/6/060301/pdf) is a paper that has a physics style proof, and [here](https://arxiv.org/abs/1603.07657) is one that derives it from the exponential generating function of $H\_n(x)H\_n(y)$.


