Chebyshev equation

Chebyshev's equation is the second order linear differential equation

${\displaystyle (1-x^{2}){d^{2}y \over dx^{2}}-x{dy \over dx}+p^{2}y=0}$

where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev.

The solutions can be obtained by power series:

${\displaystyle y=\sum _{n=0}^{\infty }a_{n}x^{n}}$

where the coefficients obey the recurrence relation

${\displaystyle a_{n+2}={(n-p)(n+p) \over (n+1)(n+2)}a_{n}.}$

The series converges for ${\displaystyle |x|<1}$ (note, x may be complex), 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

${\displaystyle F(x)=1-{\frac {p^{2}}{2!}}x^{2}+{\frac {(p-2)p^{2}(p+2)}{4!}}x^{4}-{\frac {(p-4)(p-2)p^{2}(p+2)(p+4)}{6!}}x^{6}+\cdots }$

and

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

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

The general solution is any linear combination of these two.

When p is a non-negative 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 polynomial of degree p and it is proportional to the Chebyshev polynomial of the first kind

${\displaystyle T_{p}(x)=(-1)^{p/2}\ F(x)\,}$ if p is even

${\displaystyle T_{p}(x)=(-1)^{(p-1)/2}\ p\ G(x)\,}$ if p is odd

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