I need help with this demonstration, i need to demonstrate that the generating function of Hermite is:
$$e^{2xt-t^2}=\sum H_p(x) \frac{t^p} {p!}$$
First of all i construct a diferential operator that if i apply it on the rigth side of the equation it turns on the Hermite differential equation (to make it zero) $$L=\left ( \frac{\partial^2}{\partial x^2}-2x\frac{\partial}{\partial x}+2t\frac{\partial }{\partial t}\right )$$ $$L\left ( \sum H_p(x) \frac{t^p} {p!}\right )=\sum \left ( H_p^{''}-2xH_p^{'}+2pH_p \right )\frac{t^p}{p!}=0$$ Now i apply it on the other side of the equation $$L\left ( e^{2xt-t^2}\right )=0$$ If we derive we obtain zero in both sides. My doubt is, what i need to do next, because, i dont know if i only need this reasoning to afirm that is the generating function. If someone can help me with the reasoning having that equations, that finish the proof of the generating function.
Thanks!!