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ID: INPUT{id.yaml}

Title: >
  Gegenbauer polynomials
  
Definition: >
  The Gegenbauer polynomials $C_n^{\alpha}$, $n\geq 0$, 
  can be defined via the recurrence formula CITE{formula-recurrence}.
  
Parameters:
  alpha:
    display: $\alpha$
    type: R
  n:
    type: Z
    constraints: $n \geq 0$ 
         
Comments:
  comment-orthogonal: >
    The $C_n^{\alpha}$ are orthogonal with respect to the inner product
    $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x^2)^{\alpha-1/2} dx$.
  
Formulas: 
  formula-recurrence: >
    $C_0^{\alpha}(x) = 1$, 
    $C_1^{\alpha}(x) = 2\alpha x$, 
    and $(n+1)C_{n+1}^{\alpha}(x) = 
     2x(n+\alpha) C_n^{\alpha}(x) - (n+2\alpha-1) C_{n-1}^{\alpha}(x)$ 
    for $n\geq 1$ (recurrence formula).
  forumla-generating-function: >
    $\sum_{n=0}^\infty C_n^{\alpha}(x)t^n = \frac{1}{(1-2tx+t^2)^\alpha}$ 
    (generating function).
    
Programs: 
  program-sage:
    language: Sage
    code: |
      polynomials = {alpha: 
        {n: gegenbauer(n,alpha,x) for n in [0..10]}
        for alpha in [-1/2,0,1/2,1]
      }

References:

Links:
  MathWorld:
    title: "MathWorld: Gegenbauer polynomials"
    url: https://mathworld.wolfram.com/GegenbauerPolynomial.html
  Wiki:
    title: "Wikipedia: Gegenbauer polynomials"
    url: https://en.wikipedia.org/wiki/Gegenbauer_polynomials

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

Data properties:
  type: Q[]
  complete: no
  
Display properties:
  number-header: $C_n^{\alpha}(x)$

Data: INPUT{polynomials.yaml}

— preview —

Gegenbauer polynomials

edit on github

Polynomials

$\alpha$

$n$

$C_n^{\alpha}(x)$

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

Definition

The Gegenbauer polynomials $C_n^{\alpha}$, $n\geq 0$, can be defined via the recurrence formula (1).

Parameters

$\alpha$

— real number

$n$

— integer ($n \geq 0$)

Formulas

(1)

$C_0^{\alpha}(x) = 1$, $C_1^{\alpha}(x) = 2\alpha x$, and $(n+1)C_{n+1}^{\alpha}(x) =
2x(n+\alpha) C_n^{\alpha}(x) - (n+2\alpha-1) C_{n-1}^{\alpha}(x)$
for $n\geq 1$ (recurrence formula).

(2)

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

Comments

(3)

The $C_n^{\alpha}$ are orthogonal with respect to the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)g(x) (1-x^2)^{\alpha-1/2} dx$.

Programs

(P1)

Sage

polynomials = {alpha: 
  {n: gegenbauer(n,alpha,x) for n in [0..10]}
  for alpha in [-1/2,0,1/2,1]
}

Links

[1]

MathWorld: Gegenbauer polynomials

[2]

Wikipedia: Gegenbauer polynomials

Data properties

Entries are of type: rational polynomial

Table is complete: no