Chebyshev's equation is the second order linear differential equation
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:
where the coefficients obey the recurrence relation
The series converges for (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 a<sub>0</sub> and a<sub>1</sub>, leading to the two-dimensional space of solutions that arises from second-order differential equations. The standard choices are:
and
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