orthopoly - Maple Help

orthopoly

 U
 Chebyshev polynomial of the second kind

 Calling Sequence U(n, x)

Parameters

 n - non-negative integer x - algebraic expression

Description

 • The U(n, x) function computes the nth Chebyshev polynomial of the second kind evaluated at x.
 • These polynomials are orthogonal on the interval $\left[-1,1\right]$ with respect to the weight function $w\left(x\right)=\sqrt{-{x}^{2}+1}$. They satisfy:

${\int }_{-1}^{1}w\left(t\right)U\left(m,t\right)U\left(n,t\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{\mathrm{\pi }}{2}& n=m\end{array}\right\$

 • Chebyshev polynomials of the second kind satisfy the following recurrence relation:

$U\left(0,x\right)=1,$

$U\left(1,x\right)=2x,$

$U\left(n,x\right)=2xU\left(n-1,x\right)-U\left(n-2,x\right),\mathrm{for n>1.}$

Examples

 > $\mathrm{with}\left(\mathrm{orthopoly}\right):$
 > $U\left(2,x\right)$
 ${4}{}{{x}}^{{2}}{-}{1}$ (1)
 > $U\left(3,x\right)$
 ${8}{}{{x}}^{{3}}{-}{4}{}{x}$ (2)
 > $U\left(50,\frac{1}{3}\right)$
 ${-}\frac{{40279025819534204641685}}{{717897987691852588770249}}$ (3)