A reciprocal relation for Hermite polynomials

Abstract

For $x\in \mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be written
\begin{eqnarray*}
\displaystyle
H_k(x)= \mathbb{E} \left[ (x + {\rm i} N)^k \right] =\sum_{j=0}^k {k\choose j} x^{k-j} {\rm i}^j \mathbb{E} \left[ N^j \right],
\end{eqnarray*}
where ${\rm i} = \sqrt{-1}$ and $N$ is a unit normal random variable.  We prove the reciprocal relation
\begin{eqnarray*}
\displaystyle
x^k=\sum_{j=0}^k {k\choose j} H_{k-j}(x)\ \mathbb{E} \left[ N^j \right].
\end{eqnarray*}
A similar result is given for the multivariate Hermite polynomial.