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 $\left[\mathrm{-1}\,1\right]$ with respect to the weight function $w\left(x\right)=\sqrt{-x^2+1}$. They satisfy:

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

where ChebyshevU(0,x) = 1 and ChebyshevU(1,x) = 2*x.

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

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

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

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

$8x^3-4x$

$\mathrm{ChebyshevU}\left(3.2\,2.1\right)$

$86.44386715$