ChebyshevU - Maple Programming Help

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ChebyshevU

Chebyshev function of the second kind

 Calling Sequence ChebyshevU(n, x)

Parameters

 n - algebraic expression (the degree) x - algebraic expression

Description

 • If the first parameter is a non-negative integer, then the ChebyshevU(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:

${\int }_{-1}^{1}w\left(t\right)\mathrm{ChebyshevU}\left(m,t\right)\mathrm{ChebyshevU}\left(n,t\right)ⅆt=\left\{\begin{array}{cc}0& n\ne m\\ \frac{1}{2}\mathrm{\pi }& n=m\end{array}$

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

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

 where ChebyshevU(0,x) = 1 and ChebyshevU(1,x) = 2*x.
 • This definition is analytically extended for arbitrary values of the first argument by

$\mathrm{ChebyshevU}\left(n,x\right)=\left(n+1\right)\mathrm{hypergeom}\left(\left[-n,n+2\right],\left[\frac{3}{2}\right],\frac{1}{2}-\frac{x}{2}\right)$

Examples

 > $\mathrm{ChebyshevU}\left(3,x\right)$
 ${\mathrm{ChebyshevU}}{}\left({3}{,}{x}\right)$ (1)
 > $\mathrm{simplify}\left(,'\mathrm{ChebyshevU}'\right)$
 ${8}{}{{x}}^{{3}}{-}{4}{}{x}$ (2)
 > $\mathrm{ChebyshevU}\left(3.2,2.1\right)$
 ${86.44386715}$ (3)