Mathematical physics – Is this Hermite polynomial identity known?

In some problems related to physics, I discovered the curious identity
$$ sum limits_ {n_1 + n_2 + n_3 = n} frac {n!} {n_1! , n_2! , n_3!} , H_ {2n_1} (x) , H_ {2n_2} (y) , H_ {2n_3} (z) = frac {H_ {2n + 1} (r)} {2r}, $$
or $ H_n (x) = (- 1) ^ ne ^ {x ^ 2} frac {d ^ n} {dx ^ n} e ^ {- x ^ 2} $ are the Hermite polynomials and
$ r = sqrt {x ^ 2 + y ^ 2 + z ^ 2} $. Is this identity known?