The Chebyshev equation is a linear homogeneous differential equation of the second order
and is a particular case of a hypergeometric equation.
The fundamental system of solving the Chebyshev equation in the interval -1 < x < 1 for α = n2, where n is a natural number, consists of Chebyshev polynomials of the first kind of the degree n Tn(x) = cos (n arccos x) and the functions Un(x) = sin (n arccos x) related to Chebyshev polynomials of the second kind 1/n + 1 Tn + 1'(x). Chebyshev polynomials of the first kind Tn(x) serve as an effective solution to Chebyshev equation with α = n2 and on the entire real axis.
Chebyshev polynomials are a sequence of eigenfunctions for a certain Sturm-Liouville's problem, from whence their orthogonality follows. Chebyshev polynomials have the completeness property on the interval [ -1,1 ]. In this case, any continuous function which satisfies a Lipschitz condition can be expanded into a Fourier-Chebyshev series , uniformly converging on this interval. are the orthonormal Chebyshev polynomials , . The coefficient of this series is defined as: