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ID: INPUT{id.yaml}

Title: >
  Laguerre polynomials
  
Definition: >
  The Laguerre polynomials $L_n$, $n\geq 0$, 
  can be defined as the system of orthogonal polynomials 
  with respect to the inner product CITE{comment-orthogonal}
  such that $\deg L_n=n$ and $L_n(0)=1$.
  
Parameters:
  n:
    type: Z
    constraints: $n \geq 0$ 
         
Comments:
  comment-orthogonal: >
    The $L_n$ are orthogonal with respect to the inner product
    $\langle f,g\rangle = \int_{0}^{\infty} f(x)g(x) e^{-x} dx$.
  
Formulas: 
  formula-recurrence: >
    $L_0(x) = 1$, 
    $L_1(x) = 1-x$, 
    and $(n+1)L_{n+1}(x) = (2n+1-x) L_n(x) - n L_{n-1}(x)$ 
    for $n\geq 1$ (recurrence formula).
  formula-closed-form: >
    $L_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!} x^k$
    (closed form).
  forumla-generating-function: >
    $\sum_{n=0}^\infty L_n(x)t^n = \frac{1}{1-t} e^{-tx/(1-t)}$ 
    (generating function)..
  formula-differential-equation: >
    $L_n$ is a solution of Laguerre's equation
    $xy'' + (1-x)y' + ny = 0$.
    
Programs: 
  program-sage:
    language: Sage
    code: |
      polynomials = {n: laguerre(n,x) for n in [0..100]}

References:

Links:
  Wiki:
    title: "Wikipedia: Laguerre polynomials"
    url: https://en.wikipedia.org/wiki/Laguerre_polynomials

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

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

Data: INPUT{polynomials.yaml}

— preview —

Laguerre polynomials

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Polynomials

$n$

$L_n(x)$

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

Definition

The Laguerre polynomials $L_n$, $n\geq 0$, can be defined as the system of orthogonal polynomials with respect to the inner product (5) such that $\deg L_n=n$ and $L_n(0)=1$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$L_0(x) = 1$, $L_1(x) = 1-x$, and $(n+1)L_{n+1}(x) = (2n+1-x) L_n(x) - n L_{n-1}(x)$ for $n\geq 1$ (recurrence formula).

(2)

$L_n(x) = \sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{k!} x^k$ (closed form).

(3)

$\sum_{n=0}^\infty L_n(x)t^n = \frac{1}{1-t} e^{-tx/(1-t)}$ (generating function)..

(4)

$L_n$ is a solution of Laguerre's equation $xy'' + (1-x)y' + ny = 0$.

Comments

(5)

The $L_n$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{0}^{\infty} f(x)g(x) e^{-x} dx$.

Programs

(P1)

Sage

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

Links

[1]

Wikipedia: Laguerre polynomials

Data properties

Entries are of type: rational polynomial

Table is complete: no