Chebyshev equation

Chebyshev's equation is the second order linear differential equation

(1-x^{2}){d^{2}y \over dx^{2}}-x{dy \over dx}+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=\sum _{{n=0}}^{\infty }a_{n}x^{n}

where the coefficients obey the recurrence relation

a_{{n+2}}={(n-p)(n+p) \over (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-{\frac {p^{2}}{2!}}x^{2}+{\frac {(p-2)p^{2}(p+2)}{4!}}x^{4}-{\frac {(p-4)(p-2)p^{2}(p+2)(p+4)}{6!}}x^{6}+\cdots

and

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

G(x)=x-{\frac {(p-1)(p+1)}{3!}}x^{3}+{\frac {(p-3)(p-1)(p+1)(p+3)}{5!}}x^{5}-\cdots .

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.