Hermite polynomial $H_n$ is defined by $$H_n(x) := \frac{(-1)^n}{n!}e^{x^2/2} \frac{d^n}{dx^n}e^{-x^2/2} $$ and its generating function is given by $F(t,x) := \exp{(tx-t^2/2)} $, i.e. $$\sum_{n=0}^\infty t^nH_n(x) = F(t,x). $$ Then, I want to differentiate above equation with respect to $t$ and $x$ (interchanging summation and differentiation) to derive the recurrence formula $$H'_n(x) = H_{n-1}(x);\quad (n+1)H_{n+1}(x)=xH_n(x)-H_{n-1}(x) .$$
First, consider differentiating with respect to $x$. For this, how do we check the convergence of the series $$\sum_{n=0}^\infty t^nH'_n(x)$$ is uniform with resect to $x$ (on compact sets) ?
Second, differentiate with respect to $t$. How do we compute the radius of convergence of the power series or find a dominant series?