By making use of the generating function $$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that $$(*)\quad \sum_{m,n=0}^\infty \frac{u^m}{m!} \frac{v^n}{n!} H_m(x+\frac{y}{2}) H_n(x-\frac{y}{2}) =e^{-(u^2+v^2) + 2x(u+v)+y(u-v)} .$$

I am looking for a reference which I find the analog of $(*)$ for the complex Hermite polynomials, i.e., a formula as follows $$\sum_{??}\, ?? \, H_{m,n}(z,\bar z) H_{k,l}(w,\bar w) = ?? .$$

Thank you in advance