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The U(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:

$\mathrm{with}\left(\mathrm{orthopoly}\right)\:$

$U\left(2\,x\right)$

$4x^2-1$

$U\left(3\,x\right)$

$8x^3-4x$

$U\left(50\,\frac{1}{3}\right)$

$-\frac{40279025819534204641685}{717897987691852588770249}$