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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}
