Binomial Coefficients Involving Infinite Powers of Primes

If p is a prime and n a positive integer, let νp(n) denote the exponent of p in n, and up(n) = n/p νp(n) the unit part of n. If α is a positive integer not divisible by p, we show that the p-adic limit of (−1) up((αp)!) as e → ∞ is a well-defined p-adic integer, which we call zα,p. In terms of these, we then give a formula for the p-adic limit of ( ap+c bpe+d ) as e → ∞, which we call ( ap∞+c bp∞+d ) . Here a ≥ b are positive integers, and c and d are integers. 1. Statement of results Let p be a prime number, fixed throughout. The set Zp of p-adic integers consists of expressions of the form x = ∞ ∑ i=0 cip i with 0 ≤ ci ≤ p−1. The nonnegative integers are those x for which the sum is finite. The metric on Zp is defined by d(x, y) = 1/pν(x−y), where ν(x) = min{i : ci 6= 0}. (See, e.g., [3].) The prime p will be implicit in most of our notation. If n is a positive integer, let u(n) = n/p denote the unit factor of n (with respect to p). Our first result is Theorem 1.1. Let α be a positive integer which is not divisible by p. If p > 4, then u((αpe−1)!) ≡ (−1) u((αp)!) mod p. Corollary 1.2. If α is as in Theorem 1.1, then lim e→∞ (−1) u((αp)!) exists in Zp. We denote this limiting p-adic integer by zα. If p = 2 or α is even, then zα could be thought of as u((αp ∞)!). It is easy for Maple to compute zα mod p m for m fairly large. For example, if p = 2, then z1 ≡ 1+2+2+ Date: January 29, 2013.