:
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
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).