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ID: INPUT{id.yaml}

Title: >
  Hermite polynomials in probabilist's convention
  
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:
    type: Z
    constraints: $n \geq 0$ 
         
Comments:
  comment-orthogonal: >
    The $He_n$ are orthogonal with respect to the inner product
    $\langle f,g\rangle = \int_{-\infty}^{\infty} f(x)g(x) e^{-x^2/2} dx$.
  comment-standardization: > 
    $He_n$ relates to the 
    HREF{Hermite_polynomials_H}[Hermite polynomials in physicist's convention]
    $H_n$ 
    via $He_n(x) = 2^{-n/2} H_n(x/\sqrt{2})$.
  
Formulas: 
  formula-recurrence: >
    $He_0(x) = 1$, 
    $He_1(x) = x$, 
    and $He_{n+1}(x) = x He_n(x) - He_{n}'(x)$ 
    for $n\geq 1$ (recurrence formula).
  formula-closed-form: >
    $He_n(x) = n!\sum_{k=0}^{[n/2]} \frac{(-1)^k}{k!(n-2k)!2^k} x^{n-2k}$
    (closed form).
  forumla-generating-function: >
    $\sum_{n=0}^\infty He_n(x)\frac{t^n}{n!} = e^{xt-t^2/2}$ 
    (exponential generating function).
    
Programs: 
  program-sage:
    language: Sage
    code: |
      polynomials = {n: (2^(-n/2)*hermite(n,x/sqrt(2))).expand() 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: $He_n(x)$

Data: INPUT{polynomials.yaml}

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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

[1]

Wikipedia: Hermite polynomials

Data properties

Entries are of type: integral polynomial

Table is complete: no