My question concerns an addition formula that can be found on the Wikipedia page of Hermite Polynomials but I can't find it anywhere else. The well-known formula that can be found in many books is the following
$$
\begin{aligned}
H_{n}(x+y) &=2^{-{\frac {n}{2}}}\cdot \sum _{k=0}^{n}{\binom {n}{k}}H_{n-k}\left(x{\sqrt {2}}\right)H_{k}\left(y{\sqrt {2}}\right)
\end{aligned}
$$
The formula I seek a reference for is the following
$$
H_{n}(x+y) =\sum _{k=0}^{n}{\binom {n}{k}}H_{k}(x)(2y)^{(n-k)}
$$
Can you please point me to a reference where I can find it.
Best Answer
We have the generating function for Hermite polynomials \begin{eqnarray*} \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}=e^{2xt-t^2}. \end{eqnarray*} So \begin{eqnarray*} \sum_{n=0}^{\infty} H_n(x+y) \frac{t^n}{n!}&=&e^{2yt}e^{2xt-t^2} \\ &=& \sum_{i=0}^{\infty} \frac{(2yt)^i}{i!}\sum_{k=0}^{\infty} H_k(x) \frac{t^k}{k!} \end{eqnarray*} Now collect the $t^n$ terms and we have the result \begin{eqnarray*} H_n(x+y)=\sum_{k=0}^{n} \binom{n}{k} H_k(x) (2y)^{n-k}. \end{eqnarray*}