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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 $\langle f,g\rangle = \int_{-\infty}^{\infty} f(x)g(x) e^{-x^2/2} dx$.
(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
Data properties
Entries are of type: integral polynomial
Table is complete: no