I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":
$$ \begin{align*} H_{0}(x) &=1\\ H_{1}(x) &=x\\ H_{2}(x) &= x^{2} - 1 \\ H_{3}(x) &= x^{3} - 3 x \\ & \vdots \end{align*} $$.
I found an algorithm on Wikipedia for generating Hermite polynomials using the recurrence relation $$ \begin{equation} a_{n+1,k} = \begin{cases} -na_{n-1,k} \quad k =0 \\ a_{n,k-1} - na_{n-1,k} \quad k>0 \end{cases} \end{equation} $$ where $a_{n,k}$ is the $k^{\text{th}}$ coefficient for the $n^{\text{th}}$ Hermite polynomial starting with $a_{0,0}=1$, $a_{1,0}=0$, and $a_{1,1}=1$.
My problem is that I am having a hard time finding a proof of this algorithm or at least a description of how the algorithm is derived. I am mostly looking for a source to cite in my paper, but if someone who knows a strategy to prove this, maybe starting with recurrence relation $H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)$? Any help would is appreciated.
