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Chebyshev equation

From Encyclopedia of Mathematics
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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.

How to Cite This Entry:
Chebyshev equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_equation&oldid=43461
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article