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Hermite polynomials in probabilist's convention

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Polynomials

$n$

$He_n(x)$

INPUT{polynomials.yaml} (not shown in preview)

Definition

The Hermite polynomials $He_n$, $n\geq 0$, in the probabilist's convention, can be defined as $He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$He_0(x) = 1$, $He_1(x) = x$, and $He_{n+1}(x) = x He_n(x) - He_{n}'(x)$ for $n\geq 1$ (recurrence formula).

(2)

$He_n(x) = n!\sum_{k=0}^{[n/2]} \frac{(-1)^k}{k!(n-2k)!2^k} x^{n-2k}$ (closed form).

(3)

$\sum_{n=0}^\infty He_n(x)\frac{t^n}{n!} = e^{xt-t^2/2}$ (exponential generating function).

Comments

(4)

The $He_n$ 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)

$He_n$ relates to the Hermite polynomials in physicist's convention $H_n$ via $He_n(x) = 2^{-n/2} H_n(x/\sqrt{2})$.

Programs

(P1)

Sage

```
polynomials = {n: (2^(-n/2)*hermite(n,x/sqrt(2))).expand() for n in [0..100]}
```

Links

Data properties

Entries are of type: integral polynomial

Table is complete: no