2013/01/26 by Donald M. Davis, Davis, Donald M. · 1 citation
Mathematics · #05A10 #11B65 #11D88 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11B65 #msc:11D88
paper · pdf · doi:10.48550/arxiv.1301.6285
5 pages
arxiv created 2013/01/26 · arxiv updated 2013/01/29
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/pv(n) the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)pke u((kpe)!) as e goes to infinity is a well-defined p-adic integer, which we call zk. In terms of these, we give a formula for the p-adic limit of binoma pe +c, b pe +d) as e goes to infinity, which we call binom(a p^∞ +c, b p^∞ +d). Here a ≥ b are positive integers, and c and d are integers.