Chebyshev equation

The linear homogeneous second-order ordinary differential equation

or, in self-adjoint form,

where is a constant. Chebyshev's equation is a special case of the hypergeometric equation.

The points and are regular singular points (cf. Regular singular point) of the Chebyshev equation. Substituting the independent variable

reduces this equation to a corresponding linear equation with constant coefficients:

so that Chebyshev's equation can be integrated in closed form. A fundamental systems of solutions to Chebyshev's equation on the interval with , where is a natural number, consists of the Chebyshev polynomials (of the first kind) of degree ,

and the functions , which are related to Chebyshev polynomials of the second kind. The polynomial is a real solution to Chebyshev's equation on the entire real line, with . Chebyshev's equation is also studied in complex domains.