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ID: INPUT{id.yaml}

Title: >
  Legendre polynomials
  
Definition: >
  The Legendre polynomials $P_n$, $n\geq 0$, 
  can be defined as the system of orthogonal polynomials 
  with respect to the inner product CITE{comment-orthogonal}
  such that $\deg P_n=n$ and $P_n(1)=1$.
  
Parameters:
  n:
    type: Z
    constraints: $n \geq 0$ 
         
Comments:
  comment-orthogonal: >
    The $P_n$ are orthogonal with respect to the inner product
    $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) dx$.
  
Formulas: 
  formula-recurrence: >
    $P_0(x) = 1$, 
    $P_1(x) = x$, 
    and $(n+1)P_{n+1}(x) = (2n+1)_x P_n(x) - n P_{n-1}(x)$ 
    for $n\geq 1$ (Bonnet's recursion formula).
  forumla-generating-function: >
    $\sum_{n=0}^\infty P_n(x)t^n = \frac{1}{\sqrt{1-2tx+t^2}}$ 
    (generating function).
    
Programs: 
  program-sage:
    language: Sage
    code: |
      polynomials = {n: legendre_P(n,x) for n in [0..100]}

References:

Links:
  Wiki:
    title: "Wikipedia: Legendre polynomials"
    url: https://en.wikipedia.org/wiki/Legendre_polynomials

Similar tables:
  
Keywords:
  
Tags:
- polynomial
- orthogonal polynomials

Data properties:
  type: Q[]
  complete: no
  
Display properties:
  number-header: $P_n(x)$

Data: INPUT{polynomials.yaml}

— preview —

Legendre polynomials

edit on github

Polynomials

$n$

$P_n(x)$

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

Definition

The Legendre polynomials $P_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (3) such that $\deg P_n=n$ and $P_n(1)=1$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$P_0(x) = 1$, $P_1(x) = x$, and $(n+1)P_{n+1}(x) = (2n+1)_x P_n(x) - n P_{n-1}(x)$ for $n\geq 1$ (Bonnet's recursion formula).

(2)

$\sum_{n=0}^\infty P_n(x)t^n = \frac{1}{\sqrt{1-2tx+t^2}}$ (generating function).

Comments

(3)

The $P_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) dx$.

Programs

(P1)

Sage

polynomials = {n: legendre_P(n,x) for n in [0..100]}

Links

[1]

Wikipedia: Legendre polynomials

Data properties

Entries are of type: rational polynomial

Table is complete: no