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 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 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