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ID: INPUT{id.yaml}

Title: >
  Chebyshev polynomials of the second kind $U_n$
  
Definition: >
  The Chebyshev polynomials of the second kind $U_n$, $n\geq 0$, 
  can be defined via $U_n(\cos \alpha)\sin\alpha  = \sin((n+1)\alpha)$.

Parameters:
  n:
    type: Z
    constraints: $n \geq 0$ 
         
Comments:
  comment-orthogonal: >
    The $U_n$ are orthogonal with respect to the inner product
    $\langle f,g\rangle = \int_{-1}^1 f(x)g(x)(1-x^2)^{1/2} dx$.

Formulas: 
  formula-recurrence: >
    $U_0(x) = 1$, 
    $U_1(x) = 2x$, 
    and $U_{n+1}(x) = 2x U_n(x)-U_{n-1}(x)$ for $n\geq 1$ (recurrence).
  forumla-generating-function: >
    $\sum_{n=0}^\infty U_n(x)t^n = \frac{1}{1-2tx+t^2}$ (generating function).
    
Programs: 
  program-sage:
    language: Sage
    code: |
      polynomials = {n: chebyshev_U(n,x) for n in [0..100]}

References:

Links:
  Wiki:
    title: "Wikipedia: Chebyshev polynomials"
    url: https://en.wikipedia.org/wiki/Chebyshev_polynomials

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

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

Data: INPUT{polynomials.yaml}

— preview —

Chebyshev polynomials of the second kind $U_n$

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Polynomials

$n$

$U_n(x)$

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

Definition

The Chebyshev polynomials of the second kind $U_n$, $n\geq 0$, can be defined via $U_n(\cos \alpha)\sin\alpha = \sin((n+1)\alpha)$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$U_0(x) = 1$, $U_1(x) = 2x$, and $U_{n+1}(x) = 2x U_n(x)-U_{n-1}(x)$ for $n\geq 1$ (recurrence).

(2)

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

Comments

(3)

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

Programs

(P1)

Sage

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

Links

[1]

Wikipedia: Chebyshev polynomials

Data properties

Entries are of type: integral polynomial

Table is complete: no