A reciprocal relation for Hermite polynomials

Authors

  • Saralees Nadarajah University of Manchester
  • C Withers Callaghan Innovation

DOI:

https://doi.org/10.53733/88

Keywords:

Hermite polynomial, Multivariate normal distribution, Normal distribution

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.

 

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Published

14-12-2021

How to Cite

Nadarajah, S., & Withers, C. (2021). A reciprocal relation for Hermite polynomials. New Zealand Journal of Mathematics, 51, 109–114. https://doi.org/10.53733/88

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Section

Articles