Chebyshev equation

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Chebyshev's equation is the second order linear differential equation

where p is a real constant. The equation is named after Russian mathematician Pafnuty Chebyshev.

The solutions are obtained by power series:

where the coefficients obey the recurrence relation

These series converge for x in , as may be seen by applying the ratio test to the recurrence.

The recurrence may be started with arbitrary values of a0 and a1, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are:

a0 = 1 ; a1 = 0, leading to the solution


a0 = 0 ; a1 = 1, leading to the solution

The general solution is any linear combination of these two.

When p is an integer, one or the other of the two functions has its series terminate after a finite number of terms: F terminates if p is even, and G terminates if p is odd. In this case, that function is a pth degree polynomial (converging everywhere, of course), and that polynomial is proportional to the pth Chebyshev polynomial.

if p is even
if p is odd

This article incorporates material from Chebyshev equation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.