Chebyshev equation

Chebyshev’s equation is the second orderPlanetmathPlanetmath linear differential equation


where p is a real constant.

There are two independent solutions which are given as series by:




In each case, the coefficients are given by the recursion


with y1 arising from the choice a0=1, a1=0, and y2 arising from the choice a0=0, a1=1.

The series convergePlanetmathPlanetmath for |x|<1; this is easy to see from the ratio testMathworldPlanetmath and the recursion formulaMathworldPlanetmath above.

When p is a non-negative integer, one of these series will terminate, giving a polynomial solution. If p0 is even, then the series for y1 terminates at xp. If p is odd, then the series for y2 terminates at xp.

These polynomials are, up to multiplication by a constant, the Chebyshev polynomialsDlmfPlanetmath. These are the only polynomial solutions of the Chebyshev equation.

(In fact, polynomial solutions are also obtained when p is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of p by -p.)

Title Chebyshev equation
Canonical name ChebyshevEquation
Date of creation 2013-03-22 13:10:17
Last modified on 2013-03-22 13:10:17
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 34A30
Synonym Chebyshev differential equation
Related topic HermiteEquation