Chebyshev equation
There are two independent solutions which are given as series by:
and
In each case, the coefficients are given by the recursion
with arising from the choice , , and arising from the choice , .
The series converge for ; this is easy to see from the ratio test and the recursion formula above.
When is a non-negative integer, one of these series will terminate, giving a polynomial solution. If is even, then the series for terminates at . If is odd, then the series for terminates at .
These polynomials are, up to multiplication by a constant, the Chebyshev polynomials. These are the only polynomial solutions of the Chebyshev equation.
(In fact, polynomial solutions are also obtained when is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of by .)
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 |