Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 20

In the physicists' Hermite polynomial convention, begin by differentiating the Gaussian function:The defining prefactor then givesThe derivative-based definition uses nonnegative integers; negative orders are not part of this polynomial definition.

Let $u=e^{-x^2}$, so $u^{\prime }+2xu=0$. Differentiate this identity $n$ times, for $n\ge 1$. By the Leibniz rule, only the first two derivatives of the linear factor $x$ can contribute:Rearranging provesThe case $n=0$ is the starting identity $u^{\prime }+2xu=0$, treated separately without introducing a negative derivative order.

Differentiate the product in the definition of $H_n$. Using the identity in (b),The last expression is $2n{H}_{n-1}$, soAlso $H_0^{\prime }=0$ separately. This is the Hermite polynomial derivative recurrence in the physicists' normalization.