Chebyshev’s equation is the second order linear differential equation

where $p$ is a real constant.

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

and

In each case, the coefficients are given by the recursion

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 $p\geq 0$ 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 $-p$.)