Physlib.Mathematics.SpecialFunctions.PhysHermite

The physicist's Hermite polynomials $H_n(x)$ are a sequence of polynomials with integer coefficients (elements of $\mathbb{Z}[x]$) defined recursively for $n \in \mathbb{N}$ by the base case $$H_0(x) = 1$$ and the recurrence relation $$H_{n+1}(x) = 2x H_n(x) - \frac{d}{dx} H_n(x)$$ where $x$ represents the indeterminate and $\frac{d}{dx}$ denotes the polynomial derivative.

For any natural number $n$, the physicist's Hermite polynomial $H_{n+1}(x)$ satisfies the recurrence relation $$H_{n+1}(x) = 2x H_n(x) - \frac{d}{dx} H_n(x)$$ where $H_n(x)$ denotes the $n$-th physicist's Hermite polynomial and $\frac{d}{dx}$ denotes the polynomial derivative.

For any natural number $n$, the $n$-th physicist's Hermite polynomial $H_n(x)$ is equal to the $n$-fold iteration of the operator $p(x) \mapsto 2xp(x) - \frac{d}{dx}p(x)$ applied to the constant polynomial $1$. That is, $$H_n(x) = \left( 2x - \frac{d}{dx} \right)^n (1)$$ where $x$ represents the indeterminate and $\frac{d}{dx}$ denotes the polynomial derivative.

The 0th physicist's Hermite polynomial $H_0(x)$ is equal to the constant polynomial $1$. That is, $H_0(x) = 1$.

The first physicist's Hermite polynomial $H_1(x)$ is equal to $2x$, where $x$ represents the indeterminate.

For any natural number $n$, the derivative of the $(n+1)$-th physicist's Hermite polynomial $H_{n+1}(x)$ is equal to $2(n+1) H_n(x)$. That is, $$\frac{d}{dx} H_{n+1}(x) = 2(n+1) H_n(x).$$

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The derivative of $H_n(x)$ with respect to $x$ is given by $$\frac{d}{dx} H_n(x) = 2n H_{n-1}(x),$$ where $n-1$ is defined to be $0$ if $n=0$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The sequence of polynomials satisfies the three-term recurrence relation: $$H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)$$ where $n-1$ is defined to be $0$ if $n=0$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The constant term (the coefficient of $x^0$) of $H_{n+1}(x)$ is equal to the negative of the coefficient of $x^1$ in $H_n(x)$. This can be expressed as $$[x^0] H_{n+1}(x) = -[x^1] H_n(x)$$ where $[x^k] P(x)$ denotes the coefficient of the $k$-th power of the indeterminate $x$ in the polynomial $P(x)$.

For any natural numbers $n$ and $k$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The coefficient of $x^{k+1}$ in the polynomial $H_{n+1}(x)$ is given by the relation $$[x^{k+1}] H_{n+1}(x) = 2 [x^k] H_n(x) - (k+2) [x^{k+2}] H_n(x)$$ where $[x^m] P(x)$ denotes the coefficient of the $m$-th power of the indeterminate $x$ in the polynomial $P(x)$.

For any natural numbers $n$ and $k$ such that $n < k$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The coefficient of $x^k$ in the polynomial $H_n(x)$ is zero. This can be expressed as $$[x^k] H_n(x) = 0$$ where $[x^k] P(x)$ denotes the coefficient of the $k$-th power of the indeterminate $x$ in the polynomial $P(x)$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The coefficient of the $x^n$ term in $H_n(x)$ is $2^n$. This can be expressed as $$[x^n] H_n(x) = 2^n$$ where $[x^n] P(x)$ denotes the coefficient of the $n$-th power of the indeterminate $x$ in the polynomial $P(x)$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The degree of the polynomial $H_n(x)$ is equal to $n$. This can be expressed as $$\deg(H_n(x)) = n$$ where $\deg(P)$ denotes the degree of the polynomial $P$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The natural degree of the polynomial $H_n(x)$ is equal to $n$. This can be expressed as: $$\text{natDegree}(H_n(x)) = n$$ where $\text{natDegree}(P)$ denotes the degree of the polynomial $P$ as a natural number.

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any natural numbers $n$ and $m$ such that $n < m$, the $m$-th derivative of $H_n(x)$ is equal to zero. This is expressed as: $$\frac{d^m}{dx^m} H_n(x) = 0$$ where $\frac{d^m}{dx^m}$ denotes the $m$-th order derivative with respect to $x$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The $n$-th derivative of $H_n(x)$ is the constant polynomial $n! \cdot 2^n$. This is expressed as: $$\frac{d^n}{dx^n} H_n(x) = n! \cdot 2^n$$ where $n!$ denotes the factorial of $n$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The leading coefficient of $H_n(x)$ is $2^n$.

For any natural number $n \in \mathbb{N}$, the $n$-th physicist's Hermite polynomial $H_n(x)$ is not the zero polynomial.

This instance allows a polynomial $p \in \mathbb{Z}[x]$ with integer coefficients to be treated as a function from the real numbers to the real numbers, mapping $x \in \mathbb{R}$ to the evaluation of the polynomial at that point, $p(x)$.

For any natural number $n \in \mathbb{N}$ and any real number $x \in \mathbb{R}$, the value of the $n$-th physicist's Hermite polynomial $H_n$ evaluated at $x$ is equal to its algebraic evaluation at $x$: $$H_n(x) = \operatorname{aeval}_x(H_n)$$ where $H_n(x)$ denotes the polynomial $H_n \in \mathbb{Z}[x]$ viewed as a function from $\mathbb{R}$ to $\mathbb{R}$ via coercion, and $\operatorname{aeval}_x$ denotes the algebraic evaluation of the polynomial at the point $x$.

For any real number $x$, the zeroth physicist's Hermite polynomial evaluated at $x$ is equal to $1$, denoted as $H_0(x) = 1$.

For any natural numbers $n, m \in \mathbb{N}$ and any real number $x \in \mathbb{R}$, let $H_n$ be the $n$-th physicist's Hermite polynomial. Then the $m$-th power of the value of $H_n$ at $x$ is equal to the evaluation of the polynomial $H_n^m$ at $x$: $$(H_n(x))^m = (H_n^m)(x)$$

For any natural number $n$, the $n$-th physicist's Hermite polynomial $H_n$, when viewed as a function from the real numbers $\mathbb{R}$ to $\mathbb{R}$, satisfies the three-term recurrence relation: $$H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)$$ where $n-1$ is defined to be $0$ if $n=0$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial evaluated at $x \in \mathbb{R}$. The sequence of functions satisfies the recurrence relation: $$H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)$$ where $n-1$ is defined to be $0$ if $n=0$.

For any natural numbers $n$ and $m$, the $m$-th iterated derivative of the $n$-th physicist's Hermite polynomial $H_n$ (viewed as a function) is equal to the evaluation of its $m$-th formal polynomial derivative. That is, $$\frac{d^m}{dx^m} H_n(x) = (H_n^{(m)})(x)$$ where $H_n^{(m)}$ denotes the $m$-th formal derivative of the polynomial $H_n$.

For any natural number $n$ and any real number $x$, the $n$-th physicist's Hermite polynomial $H_n$ is differentiable at $x$.

Let $H_n$ denote the $n$-th physicist's Hermite polynomial. For any natural numbers $n$ and $m$, the $m$-th iterated derivative of $H_n$, denoted by $\frac{d^m}{dx^m} H_n$, is differentiable at any point $x \in \mathbb{R}$.

For any natural number $n$, let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. The derivative of $H_n$ (as a function from $\mathbb{R}$ to $\mathbb{R}$) is given by $$\frac{d}{dx} H_n(x) = 2n H_{n-1}(x),$$ where $n-1$ is defined to be $0$ if $n=0$.

Let $E$ be a normed real vector space, $x \in E$, and $f: E \to \mathbb{R}$ be a function that is differentiable at $x$. For any natural number $n$, let $H_n$ denote the $n$-th physicist's Hermite polynomial. The Fréchet derivative of the composition $H_n \circ f$ at $x$ is given by $$D(H_n \circ f)(x) = 2n H_{n-1}(f(x)) \cdot Df(x),$$ where $Df(x)$ is the Fréchet derivative of $f$ at $x$, and $n-1$ is defined to be $0$ if $n=0$.

Let $x$ be a real number and $f: \mathbb{R} \to \mathbb{R}$ be a function that is differentiable at $x$. For any natural number $n$, let $H_n$ denote the $n$-th physicist's Hermite polynomial. The derivative of the composition $H_n \circ f$ evaluated at $x$ is given by $$\frac{d}{dx} H_n(f(x)) = 2n H_{n-1}(f(x)) f'(x),$$ where $f'(x)$ is the derivative of $f$ at $x$, and $n-1$ is defined to be $0$ if $n=0$.

For any natural number $n$ and any real number $x$, the $n$-th physicist's Hermite polynomial $H_n$ satisfies the parity relation: $$H_n(-x) = (-1)^n H_n(x)$$

For any natural number $n$ and any real number $x$, the $n$-th derivative of the Gaussian function $e^{-x^2}$ evaluated at $x$ is given by $$\frac{d^n}{dx^n} e^{-x^2} = (-1)^n H_n(x) e^{-x^2},$$ where $H_n(x)$ denotes the $n$-th physicist's Hermite polynomial.

For any natural number $n$ and any real number $x$, the $n$-th physicist's Hermite polynomial $H_n(x)$ is given by $$H_n(x) = (-1)^n \frac{\frac{d^n}{dx^n} e^{-x^2}}{e^{-x^2}},$$ where $\frac{d^n}{dx^n}$ denotes the $n$-th derivative.

For any natural number $n$ and any real number $x$, the $n$-th physicist's Hermite polynomial $H_n(x)$ is given by the formula $$H_n(x) = (-1)^n \left( \frac{d^n}{dx^n} e^{-x^2} \right) e^{x^2},$$ where $\frac{d^n}{dx^n}$ denotes the $n$-th derivative with respect to $x$.

Let $b \in \mathbb{R}$ be a positive real number ($b > 0$), and let $P \in \mathbb{Z}[x]$ be a polynomial with integer coefficients. Then the function $x \mapsto P(x) e^{-bx^2}$ is integrable over the real line $\mathbb{R}$ with respect to the Lebesgue measure.

Let $b$ and $c$ be real numbers such that $b > 0$, and let $P$ be a polynomial with integer coefficients. Then the function $f(x) = P(cx) e^{-bx^2}$ is integrable on $\mathbb{R}$ with respect to the Lebesgue measure.

For any natural numbers $n, p,$ and $m$, the product of the $m$-th derivative of the $p$-th physicist's Hermite polynomial $H_p(x)$ and the $n$-th derivative of the Gaussian function $e^{-x^2}$ is integrable over the real line $\mathbb{R}$ with respect to the Lebesgue measure. That is, the function $$f(x) = \left( \frac{d^m}{dx^m} H_p(x) \right) \left( \frac{d^n}{dx^n} e^{-x^2} \right)$$ is integrable.

For any natural numbers $n$ and $m$, the integral over the real line of the product of the $n$-th and $m$-th physicist's Hermite polynomials, $H_n(x)$ and $H_m(x)$, weighted by the Gaussian function $e^{-x^2}$, satisfies the identity: $$\int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = (-1)^m \int_{-\infty}^{\infty} H_n(x) \frac{d^m}{dx^m} (e^{-x^2}) \, dx$$ where $\frac{d^m}{dx^m}$ denotes the $m$-th derivative with respect to $x$.

For any natural numbers $n$ and $m$, and for any natural number $p$ such that $p \leq m$, the integral over the real line of the product of the $n$-th and $m$-th physicist's Hermite polynomials $H_n(x)$ and $H_m(x)$ weighted by the Gaussian function $e^{-x^2}$ satisfies the identity: $$\int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = (-1)^{m-p} \int_{-\infty}^{\infty} \left( \frac{d^p}{dx^p} H_n(x) \right) \left( \frac{d^{m-p}}{dx^{m-p}} e^{-x^2} \right) \, dx$$ where $\frac{d^k}{dx^k}$ denotes the $k$-th iterated derivative with respect to $x$.

For any natural numbers $n$ and $m$, the integral over the real line of the product of the $n$-th and $m$-th physicist's Hermite polynomials $H_n(x)$ and $H_m(x)$ weighted by the Gaussian function $e^{-x^2}$ is equal to the integral of the $m$-th derivative of $H_n(x)$ weighted by the same Gaussian function: $$\int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = \int_{-\infty}^{\infty} \left( \frac{d^m}{dx^m} H_n(x) \right) e^{-x^2} \, dx$$ where $\frac{d^m}{dx^m}$ denotes the $m$-th iterated derivative with respect to $x$.

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any natural numbers $n$ and $m$ such that $n < m$, the integral over the real line of the product of $H_n(x)$ and $H_m(x)$ weighted by the Gaussian function $e^{-x^2}$ is zero: $$\int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = 0$$

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any natural numbers $n$ and $m$ such that $n \neq m$, the integral over the real line of the product of $H_n(x)$ and $H_m(x)$ weighted by the Gaussian function $e^{-x^2}$ is zero: $$\int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = 0$$

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any natural numbers $n$ and $m$ such that $n \neq m$, and for any real number $c$, the integral over the real line of the product of $H_n(cx)$ and $H_m(cx)$ weighted by the Gaussian function $e^{-c^2 x^2}$ is zero: $$\int_{-\infty}^{\infty} H_n(cx) H_m(cx) e^{-c^2 x^2} \, dx = 0$$

For any natural number $n$, the integral over the real line of the square of the $n$-th physicist's Hermite polynomial $H_n(x)$ weighted by the Gaussian function $e^{-x^2}$ is given by: $$\int_{-\infty}^{\infty} (H_n(x))^2 e^{-x^2} \, dx = n! 2^n \sqrt{\pi}$$ where $H_n(x)$ denotes the $n$-th physicist's Hermite polynomial.

For any natural number $n$ and any real number $c$, the integral over the real line of the square of the $n$-th physicist's Hermite polynomial $H_n$ evaluated at $cx$, weighted by the Gaussian function $e^{-c^2 x^2}$, is given by: $$\int_{-\infty}^{\infty} (H_n(cx))^2 e^{-c^2 x^2} \, dx = \left| \frac{1}{c} \right| n! 2^n \sqrt{\pi}$$ where $H_n$ denotes the $n$-th physicist's Hermite polynomial.

Let $H_k(x)$ denote the $k$-th physicist's Hermite polynomial. For any polynomial $P$ with integer coefficients (i.e., $P \in \mathbb{Z}[x]$) and any natural number $n$ such that the degree of $P$ is equal to $n$, the function $P: \mathbb{R} \to \mathbb{R}$ belongs to the $\mathbb{R}$-linear span of the set of all physicist's Hermite polynomials $\{H_k : k \in \mathbb{N}\}$.

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any polynomial $P$ with integer coefficients (i.e., $P \in \mathbb{Z}[x]$), the function $P : \mathbb{R} \to \mathbb{R}$ belongs to the $\mathbb{R}$-linear span of the set of all physicist's Hermite polynomials $\{H_n : n \in \mathbb{N}\}$.

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any real constant $c \in \mathbb{R}$, the function $f(x) = \cos(cx)$ belongs to the topological closure of the $\mathbb{R}$-linear span of the set of all physicist's Hermite polynomials $\{H_n : n \in \mathbb{N}\}$.

Let $H_n(x)$ denote the $n$-th physicist's Hermite polynomial. For any natural number $n \in \mathbb{N}$, the integral over the real line of the square of $H_n(x)$ weighted by the Gaussian function $e^{-x^2}$ is given by: $$\int_{-\infty}^{\infty} (H_n(x))^2 e^{-x^2} \, dx = n! 2^n \sqrt{\pi}.$$