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ID: INPUT{id.yaml} Title: > Hermite polynomials in physicist's convention Definition: > The Hermite polynomials $H_n$, $n\geq 0$, in the physicist's convention, can be defined as $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$. Parameters: n: type: Z constraints: $n \geq 0$ Comments: comment-orthogonal: > The $H_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-\infty}^{\infty} f(x)g(x) e^{-x^2} dx$. comment-standardization: > $H_n$ relates to the HREF{Hermite_polynomials_He}[Hermite polynomials in probabilist's convention] $He_n$ via $H_n(x) = 2^{n/2} He_n(\sqrt{2}x)$. Formulas: formula-recurrence: > $H_0(x) = 1$, $H_1(x) = 2x$, and $H_{n+1}(x) = 2x H_n(x) - H_{n}'(x)$ for $n\geq 1$ (recurrence formula). formula-closed-form: > $H_n(x) = n!\sum_{k=0}^{[n/2]} \frac{(-1)^k}{k!(n-2k)!} (2x)^{n-2k}$ (closed form). forumla-generating-function: > $\sum_{n=0}^\infty H_n(x)\frac{t^n}{n!} = e^{2xt-t^2}$ (exponential generating function). Programs: program-sage: language: Sage code: | polynomials = {n: hermite(n,x) for n in [0..100]} References: Links: Wiki: title: "Wikipedia: Hermite polynomials" url: https://en.wikipedia.org/wiki/Hermite_polynomials Similar tables: Keywords: Tags: - polynomial - orthogonal polynomials Data properties: type: Z[] complete: no Display properties: number-header: $H_n(x)$ Data: INPUT{polynomials.yaml}
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Hermite polynomials in physicist's convention
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Polynomials
$n$ 
$H_n(x)$
INPUT{polynomials.yaml} (not shown in preview)
Definition
The Hermite polynomials $H_n$, $n\geq 0$, in the physicist's convention, can be defined as $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$.
Parameters
$n$
— integer ($n \geq 0$)
Formulas
(1)
$H_0(x) = 1$, $H_1(x) = 2x$, and $H_{n+1}(x) = 2x H_n(x) - H_{n}'(x)$ for $n\geq 1$ (recurrence formula).
(2)
$H_n(x) = n!\sum_{k=0}^{[n/2]} \frac{(-1)^k}{k!(n-2k)!} (2x)^{n-2k}$ (closed form).
(3)
$\sum_{n=0}^\infty H_n(x)\frac{t^n}{n!} = e^{2xt-t^2}$ (exponential generating function).
Comments
(4)
The $H_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-\infty}^{\infty} f(x)g(x) e^{-x^2} dx$.
(5)
$H_n$ relates to the
Hermite polynomials in probabilist's convention
$He_n$ via $H_n(x) = 2^{n/2} He_n(\sqrt{2}x)$.
Programs
(P1)
Sage
polynomials = {n: hermite(n,x) for n in [0..100]}
Links
[1]
Wikipedia: Hermite polynomials
Data properties
Entries are of type: integral polynomial
Table is complete: no