Chebyshev's equation is the second order linear differential equation
The solutions are obtained by power series:
where the coefficients obey the recurrence relation
These series converge for x in , 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
- a0 = 0 ; a1 = 1, leading to the solution
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 pth degree polynomial (converging everywhere, of course), and that polynomial is proportional to the pth Chebyshev polynomial.
- if p is even
- if p is odd