The monic probabilists' Hermite polynomials

The monic probabilists' Hermite polynomials are the polynomials $H_m\in \mathbb{R}[x]$ defined by $H_0=1,H_1(x)=x,x{H}_m(x)=H_{m+1}(x)+m{H}_{m-1}(x)(m\ge 1)\mathrm{.}$ The recurrence is solved for the higher polynomial, $H_{m+1}(x)=x{H}_m(x)-m{H}_{m-1}(x)$, so it determines $H_m$ uniquely by induction on $m$. Each $H_m$ is monic of degree $m$: $H_2=x^2-1$, $H_3=x^3-3x$ and $H_4=x^4-6x^2+3$; in general $H_m(x)=m!{\sum }_{j=0}^{\lfloor m\mathrm{/}2\rfloor }\frac{(-1\mathrm{/}2{)}^j{x}^{m-2j}}{j!(m-2j)!}$.

Under the AC assumption of the Gaussian-law supplier, these are the monic orthogonal polynomials for the standard normal law of Standard normal and normal laws: they form an orthogonal system for the measure $(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}dx$; orthogonality and the expansion of monomials in this system are proved separately on this page. The moments of that law are those of Moments, variance, and covariance on a probability space, and the polynomial calculus used below is that of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set. The defining property used in this batch is the recurrence; no choice principle is used.