CHEBYSHEV EQUATION

The Chebyshev equation is a linear homogeneous differential equation of the second order

(1)

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 T_n(x) = cos (n arccos x) and the functions U_n(x) = sin (n arccos x) related to Chebyshev polynomials of the second kind 1/n + 1 T_{n + 1}'(x). Chebyshev polynomials of the first kind T_n(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:

(2)