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).