Chebyshev equation

Chebyshev's equation is the second order linear differential equation

( 1 − x 2 ) d 2 y d x 2 − x d y d x + p 2 y = 0

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

The solutions are obtained by power series:

y = ∑ n = 0 ∞ a n x n

where the coefficients obey the recurrence relation

a n + 2 = ( n − p ) ( n + p ) ( n + 1 ) ( n + 2 ) a n .

These series converge for x in [ − 1 , 1 ] , 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 F ( x ) = 1 − p 2 2 ! x 2 + ( p − 2 ) p 2 ( p + 2 ) 4 ! x 4 − ( p − 4 ) ( p − 2 ) p 2 ( p + 2 ) ( p + 4 ) 6 ! x 6 + ⋯

and

a₀ = 0 ; a₁ = 1, leading to the solution G ( x ) = x − ( p − 1 ) ( p + 1 ) 3 ! x 3 + ( p − 3 ) ( p − 1 ) ( p + 1 ) ( p + 3 ) 5 ! x 5 − ⋯ .

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 p^th degree polynomial (converging everywhere, of course), and that polynomial is proportional to the p^th Chebyshev polynomial.

T p ( x ) = ( − 1 ) p / 2   F ( x ) if p is even T p ( x ) = ( − 1 ) ( p − 1 ) / 2   p   G ( x ) if p is odd

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