2006/08/14 by Zhi-Wei Sun, Sun, Zhi-Wei
Mathematics · #05A10 #11A07 #11B65 #11B68 #11B73 #11S80 #11T24 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65 #msc:11B68 #msc:11B73 #msc:11S80 #msc:11T24
paper · pdf · doi:10.48550/arxiv.math/0608328
38 pages
arxiv created 2006/08/14 · arxiv updated 2009/12/01
Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that Fp(n,r)=(-p)-[(n-1)/(p-1)]∑k=r(mod p)\binomnk(-1)k∈\Z. Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated Fp(n,r) mod p in [SW2]. In this paper, using p-adic methods we determine (Fp(m,r)-Fp(n,r))/(m-n) modulo p in terms of Bernoulli numbers, where m>0 is an integer with m\not=n and m=n (mod p(p-1)). Consequently, Fp(n,r) mod pordp(n)+1 is determined; for example, if n=n_*(mod p-1) with 0<n_*<p-2 then (Fp(pn,0))/(pn)=(n_*!)/(n_*+1)Bp-1-n_* (mod p). This yields an application to Stirling numbers of the second kind. We also study extended Fleck quotients; in particular we prove that if a>0 and l≥ 0 are integers with 2≤ n-l≤ p then \frac1pn-l∑l<k≤ n \binompa n-dpa k-d(-1)pk\binomk-1l =\frac(-1)l-1n!l!(n-l)Bp-n+l (mod p) for all d=1,...,maxpa-2,1.