We start this section by giving another proof of Lucas' Theorem (due to Fine (1947)), based on the obvious generating function for : Start by noting that as each is divisible by p, by Kummer's Theorem, unless i=0 or . Therefore, writing n in base p, we have
and the result follows.
We can use the same approach to try to prove the analogue of Lucas' Theorem modulo , and arbitrary prime powers, but the details become much more complicated than in the proof given in section 2. We may also generalize this method to evaluate, modulo p, the coefficients of powers of any given polynomial:
Given an integer polynomial of degree d, define , and let if m<0 or m>nd (note that when ). Clearly using Fermat's Theorem, and so
But if m = pt+r then r is of the form and so we obtain the following generalization of (1):
We use a similar approach in the proof of (11).