ChebyshevT - Maple Programming Help

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ChebyshevT

Chebyshev function of the first kind

 Calling Sequence ChebyshevT(n, x)

Parameters

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

Description

 • If the first parameter is a non-negative integer, the ChebyshevT(n, x) function computes the nth Chebyshev polynomial of the first kind evaluated at x.
 • These polynomials are orthogonal on the interval (-1, 1) with respect to the weight function $w\left(x\right)=\frac{1}{\sqrt{-{x}^{2}+1}}$. These polynomials satisfy the following:

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

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

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

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

$\mathrm{ChebyshevT}\left(a,x\right)=\mathrm{hypergeom}\left(\left[-a,a\right],\left[\frac{1}{2}\right],\frac{1}{2}-\frac{x}{2}\right)$

Examples

 > $\mathrm{ChebyshevT}\left(3,x\right)$
 ${\mathrm{ChebyshevT}}{}\left({3}{,}{x}\right)$ (1)
 > $\mathrm{simplify}\left(,'\mathrm{ChebyshevT}'\right)$
 ${4}{}{{x}}^{{3}}{-}{3}{}{x}$ (2)
 > $\mathrm{ChebyshevT}\left(2.2,0.5\right)$
 ${-0.6691306064}$ (3)
 > $\mathrm{ChebyshevT}\left(\frac{1}{3},x\right)$
 ${\mathrm{ChebyshevT}}{}\left(\frac{{1}}{{3}}{,}{x}\right)$ (4)
 > $\mathrm{series}\left(,'\mathrm{ChebyshevT}'\right)$
 ${\mathrm{cos}}{}\left(\frac{{\mathrm{arccos}}{}\left({x}\right)}{{3}}\right)$ (5)
 > $\frac{ⅆ}{ⅆx}\mathrm{ChebyshevT}\left(1,x\right)$
 ${-}\frac{{x}{}{\mathrm{ChebyshevT}}{}\left({1}{,}{x}\right)}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{{\mathrm{ChebyshevT}}{}\left({0}{,}{x}\right)}{{-}{{x}}^{{2}}{+}{1}}$ (6)