The function U_n(x) represents the Chebyshev polynomial of the second kind, which is a sequence of orthogonal polynomials. These polynomials are defined on the interval [-1, 1] and are used in various applications, including numerical analysis and approximation theory. The n -th polynomial can be expressed using the formula U_n(x) = \frac\sin((n+1) \theta)\sin(\theta) , where x = \cos(\theta) . Chebyshev polynomials, including U_n(x) , are particularly useful in minimizing errors in polynomial interpolation. They are closely related to the Chebyshev polynomials of the first kind, denoted as T_n(x) , and share properties that make them valuable in fields such as signal processing and computer graphics.