Chebyshev equation

Chebyshev's equation is the second order linear differential equation $$(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2y = 0$$ where $p$ is a real constant.

There are two independent solutions which are given as series by:

In each case, the coefficients are given by the recursion $$ a_{n+2} = \frac{(n-p)(n+p)}{(n+1)(n+2)} a_n $$ with $y_1$ arising from the choice $a_0 = 1$ , $a_1 = 0$ , and $y_2$ arising from the choice $a_0 = 0$ , $a_1 = 1$ .

The series converge for $|x| < 1$ ; this is easy to see from the ratio test and the recursion formula above.

When $p$ is a non-negative integer, one of these series will terminate, giving a polynomial solution. If $p \ge 0$ is even, then the series for $y_1$ terminates at $x^p$ . If $p$ is odd, then the series for $y_2$ terminates at $x^p$ .

These polynomials are, up to multiplication by a constant, the Chebyshev polynomials. These are the only polynomial solutions of the Chebyshev equation.

(In fact, polynomial solutions are also obtained when $p$ is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of $p$ by $-p$ .)