In section 11 we apply the results from here to binomial coefficients. We start by proving

* Proposition 3.* * If x is divisible by prime p then
*

**(34)**

*
where the limit is taken p-adically. *

* Proof:* Suppose that . If then
divides the numerator of both and . If and *r* is sufficiently large then

so that

Therefore, letting and then , we obtain (34).

For each , define

where the limit here is taken *p*-adically: Note that this limit
exists and is well defined by (29); moreover for all odd *n*.
(Using Theorem 5.11 of Washington's book, one can also show that
,
where is the *p*-adic *L*-function, and is that
*p*-adic st root of unity for which
.)

Our main result of this section is

* Proposition 4.* * For any integer x we have
*

**(35)**

*
where () is chosen so that
, and
*

* *

(Using Theorem 5.11 of Washington's book, one can also show that
,
where : note that has a pole at *s=1*.)

* Proof:* As whenever
*p* does not divide *j*, we have

by Proposition 3. Now, fix and take in (30), so that

for all sufficiently large *r*, as
for those *k* in the sum.
Therefore, letting and then , we obtain (35).