Chebyshev equation

Chebyshev's equation is the second order linear differential equation



(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + 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:


 * $$y = \sum_{n=0}^\infty a_nx^n$$

where the coefficients obey the recurrence relation


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

The series converges for $$|x|<1$$ (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
 * $$F(x) = 1 - \frac{p^2}{2!}x^2 + \frac{(p-2)p^2(p+2)}{4!}x^4 - \frac{(p-4)(p-2)p^2(p+2)(p+4)}{6!}x^6 + \cdots $$

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 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


 * $$T_p(x) = (-1)^{p/2}\ F(x)\,$$ if p is even
 * $$T_p(x) = (-1)^{(p-1)/2}\ p\ G(x)\,$$ if p is odd