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orthopoly

 T
 Chebyshev polynomial (of the first kind)

 Calling Sequence T(n, x)

Parameters

 n - non-negative integer x - algebraic expression

Description

 • The T(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 $w\left(x\right)=\frac{1}{\sqrt{-{x}^{2}+1}}$. They satisfy:

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

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

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

$T\left(1,x\right)=x,$

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

Examples

 > $\mathrm{with}\left(\mathrm{orthopoly}\right):$
 > $T\left(2,x\right)$
 ${2}{}{{x}}^{{2}}{-}{1}$ (1)
 > $T\left(3,x\right)$
 ${4}{}{{x}}^{{3}}{-}{3}{}{x}$ (2)
 > $T\left(50,\frac{1}{3}\right)$
 $\frac{{203160711589551869872313}}{{717897987691852588770249}}$ (3)