Hermite Polynomials with a Given Variance

In the setting of The Real Numbers: Standing Notation and Background, each natural number being identified with its image in $\mathbb{R}$ under the canonical map, and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\subseteq\mathbb{R}$. For $n\in\mathbb{N}$, $n!$ is the factorial of $n$, and $0!=1$; for real $y$ and $j\in\mathbb{N}$, $y^{j}$ is the $j$th power of $y$, and $y^{0}=1$. Sums over nonempty finite index sets are those of Sum over a Finite Index Set; in particular, for $n\in\mathbb{N}_{0}$ and real numbers $f(0),\dots,f(n)$, $\sum_{j=0}^{n}f(j)$ is the sum over the index set $\{0,1,\dots,n\}$, which equals $f(0)$ if $n=0$ and $f(0)+\sum_{j=1}^{n}f(j)$ if $n\in\mathbb{N}$.

(Hermite polynomials)¶ Let $v\in\mathbb{R}$ with $0\le v$ and let $n\in\mathbb{N}_{0}$, and put $J_{n}=\{j\in\mathbb{N}_{0}:2j\le n\}$, a finite set since it is contained in $\{0,1,\dots,n\}$, and nonempty since $0\in J_{n}$; for $j\in J_{n}$, the integer $n-2j$ satisfies $0\le n-2j\le n$, so $n-2j\in\mathbb{N}_{0}$, and $2^{j}\,j!\,(n-2j)!$ is a product of positive real numbers, hence positive. The Hermite polynomial of degree $n$ with variance $v$ is the function $H^{v}_{n}:\mathbb{R}\to\mathbb{R}$,

$$H^{v}_{n}(t)=\sum_{j\in J_{n}}\frac{n!}{2^{j}\,j!\,(n-2j)!}\,(-v)^{j}\,t^{\,n-2j}\qquad(t\in\mathbb{R}).$$