Chebyshev function of the second kind

A modified set of Chebyshev polynomials defined by a slightly different generating function. They arise in the development of four-dimensional spherical harmonics in angular momentum theory. They are a special case of the Gegenbauer polynomial with α = 1. They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the second kind are denoted U_n(x), and implemented in the Wolfram Language as ChebyshevU[n, x]. The polynomials U_n(x) are illustrated above for x element [-1, 1] and n = 1, 2, ..., 5.

Chebyshev approximation formula | Chebyshev polynomial of the first kind | Gegenbauer polynomial