Hermite Polynomials: Low Orders, Derivatives, Recursion, Scaling, the Addition Formula, Gaussian Orthogonality and the Mehler Identity

In the settings of The Real Numbers: Standing Notation and Background and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, with $\mathbb{N}_{0}$, factorials, powers, sums $\sum_{j=0}^{n}$ and the Hermite polynomials $H^{v}_{n}$ ($0\le v$, $n\in\mathbb{N}_{0}$) as in Hermite Polynomials with a Given Variance. The real line is identified with the Euclidean space $\mathbb{R}^{1}$, its Borel $\sigma$-algebra with $\mathcal{B}(\mathbb{R}^{1})$; for a function $f:\mathbb{R}\to\mathbb{R}$ of class $C^{1}$, respectively $C^{2}$, on $\mathbb{R}^{1}$, $f'$ is its partial derivative $\partial_{1}f$ and $f''$ is $\partial_{1}\partial_{1}f$, as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives with $q=1$. For $n,j\in\mathbb{N}_{0}$ with $j\le n$, $\binom{n}{j}=\frac{n!}{j!\,(n-j)!}$. For real $v>0$, $\gamma_{v}$ is the diagonal Gaussian measure on $\mathbb{R}^{1}$ with variance vector $(v)$; integrability and integrals with respect to it are those of Measure Spaces and the Lebesgue Integral: Standing Notation, and $\int_{\mathbb{R}}f(y)\,\gamma_{v}(dy)$ denotes $\int_{\mathbb{R}}f\,d\gamma_{v}$. Let $v,w\in\mathbb{R}$ with $0\le v$ and $0\le w$, and let $n\in\mathbb{N}_{0}$.

1. (Low orders)¶ For every $t\in\mathbb{R}$: $H^{v}_{0}(t)=1$, $H^{v}_{1}(t)=t$, $H^{v}_{2}(t)=t^{2}-v$, $H^{v}_{3}(t)=t^{3}-3vt$, $H^{v}_{4}(t)=t^{4}-6vt^{2}+3v^{2}$, and $H^{0}_{n}(t)=t^{n}$.

2. (Polynomial growth and regularity)¶ $H^{v}_{n}$ is a polynomial function on $\mathbb{R}$ and is of class $C^{2}$ on $\mathbb{R}^{1}$; there are $b_{0},\dots,b_{n}\in\mathbb{R}$ with $b_{n}=1$ and $H^{v}_{n}(t)=\sum_{i=0}^{n}b_{i}t^{i}$ for every $t\in\mathbb{R}$; and there is $M\in\mathbb{R}$ with $|H^{v}_{n}(t)|\le M(1+|t|^{n})$ for every $t\in\mathbb{R}$.

3. (Derivative)¶ $(H^{v}_{0})'(t)=0$ for every $t\in\mathbb{R}$, and, if $n\in\mathbb{N}$, $(H^{v}_{n})'(t)=n\,H^{v}_{n-1}(t)$ for every $t\in\mathbb{R}$.

4. (Recursion)¶ For every $t\in\mathbb{R}$, $H^{v}_{n+1}(t)=t\,H^{v}_{n}(t)-v\,(H^{v}_{n})'(t)$; in particular, if $n\in\mathbb{N}$, $H^{v}_{n+1}(t)=t\,H^{v}_{n}(t)-n\,v\,H^{v}_{n-1}(t)$.

5. (Eigenfunction)¶ For every $t\in\mathbb{R}$, $v\,(H^{v}_{n})''(t)-t\,(H^{v}_{n})'(t)=-n\,H^{v}_{n}(t)$.

6. (Scaling)¶ For all $r,t\in\mathbb{R}$, $H^{r^{2}v}_{n}(rt)=r^{n}\,H^{v}_{n}(t)$, the left-hand side being defined since $0\le r^{2}v$.

7. (Addition)¶ For all $s,t\in\mathbb{R}$,

$$H^{v+w}_{n}(s+t)=\sum_{j=0}^{n}\binom{n}{j}\,H^{v}_{j}(s)\,H^{w}_{n-j}(t).$$

8. (Gaussian orthogonality)¶ Assume $v>0$, and let $m\in\mathbb{N}_{0}$. Then $H^{v}_{m}$ and $H^{v}_{m}H^{v}_{n}$ are Borel and integrable with respect to $\gamma_{v}$, and

$$\int_{\mathbb{R}}H^{v}_{m}H^{v}_{n}\,d\gamma_{v}=\begin{cases}n!\,v^{n}&\text{if }m=n,\\ 0&\text{if }m\ne n.\end{cases}$$

In particular, taking $m=0$, $\int_{\mathbb{R}}H^{v}_{n}\,d\gamma_{v}=0$ if $n\in\mathbb{N}$.

9. (Mehler identity)¶ Assume $v>0$, and let $r,s\in\mathbb{R}$ with $r^{2}+s^{2}=1$. For every $t\in\mathbb{R}$ the function $y\mapsto H^{v}_{n}(rt+sy)$ is Borel and integrable with respect to $\gamma_{v}$, and

$$\int_{\mathbb{R}}H^{v}_{n}(rt+sy)\,\gamma_{v}(dy)=r^{n}\,H^{v}_{n}(t).$$