Fleck quotients and Bernoulli numbers

Abstract

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ 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, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0<n_*<p-2$ then $$\frac{F_p(pn,0)}{pn}=\frac{n_*!}{n_*+1}B_{p-1-n_*} (mod p).$$ 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 $l\ge 0$ are integers with $2\le n-l\le p$ then $$\frac{1}{p^{n-l}}\sum_{l<k\le n} \binom{p^a n-d}{p^a k-d}(-1)^{pk}\binom{k-1}{l} =\frac{(-1)^{l-1}n!}{l!(n-l)}B_{p-n+l} (mod p)$$ for all d=1,...,max{p^{a-2},1}.