Chebyshev equation
Chebyshev's equation is the second order linear differential equation
where p is a real 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 , 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:
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
