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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