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):

**(24)**

We use a similar approach in the proof of (11).