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:

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

where ChebyshevT(0,x) = 1 and ChebyshevT(1,x) = x.

This definition is analytically extended for arbitrary values of the first argument by

$\mathrm{ChebyshevT}\left(3\,x\right)$

$\mathrm{ChebyshevT}\left(3\,x\right)$

$\mathrm{simplify}\left(\,'\mathrm{ChebyshevT}'\right)$

$4x^3-3x$

$\mathrm{ChebyshevT}\left(2.2\,0.5\right)$

$\mathrm{-0.6691306064}$

$\mathrm{ChebyshevT}\left(\frac{1}{3}\,x\right)$

$\mathrm{ChebyshevT}\left(\frac{1}{3}\,x\right)$

$\mathrm{series}\left(\,'\mathrm{ChebyshevT}'\right)$

$\cos \left(\frac{\arccos \left(x\right)}{3}\right)$

$\mathrm{diff}\left(\mathrm{ChebyshevT}\left(1\,x\right)\,x\right)$

$-\frac{x\mathrm{ChebyshevT}\left(1\,x\right)}{-x^2+1}+\frac{\mathrm{ChebyshevT}\left(0\,x\right)}{-x^2+1}$