Suppose I've to find $H_4(0),$ where $H$ represents the Hermite polynomial. I've only been provided with the following relation :
$$e^{-t^2+2tx}=\sum_n H_n(x)\frac{t^n}{n!}$$
My first step is to substitute $x=0$ in the above relation, to get :
$$e^{-t^2}=\sum_n H_n(0)\frac{t^n}{n!}$$
Expanding the exponential in the form of a Taylor series, we get :
$$\sum_n \frac{(-1)^n t^{2n}}{n!}=\sum_n H_n(0)\frac{t^n}{n!}$$
However, I'm unable to proceed from here. Similarly, in legendre polynomial, I'm stuck, and I'm unable to proceed any further. In some problems regarding Euler polynomial however, this method seems to work.
Any help would be highly appreciated.