Chebyshev equation


Chebyshev’s equation is the second orderPlanetmathPlanetmath linear differential equation

(1-x2)d2ydx2-xdydx+p2y=0

where p is a real constant.

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

y1(x)=1-p22!x2+(p-2)p2(p+2)4!x4-(p-4)(p-2)p2(p+2)(p+4)6!x6+

and

y2(x)=x-(p-1)(p+1)3!x3+(p-3)(p-1)(p+1)(p+3)5!x5-

In each case, the coefficients are given by the recursion

an+2=(n-p)(n+p)(n+1)(n+2)an

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