Fleck quotients and Bernoulli numbers

Abstract

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p{)}^{-[(n-1)/(p-1)]}\sum _{k=r(modp)}(\genfrac{}{}{0ex}{}{n}{k})(-1{)}^k\in \mathbb{Z}.$$ 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 $m\ne n$ and $m=n(modp(p-1))$. Consequently, $F_p(n,r)$ mod $p^{or{d}_p(n)+1}$ is determined; for example, if $n=n_{\ast }(modp-1)$ with $0<n_{\ast }<p-2$ then $$\frac{F_p(pn,0)}{pn}=\frac{n_{\ast }!}{n_{\ast }+1}{B}_{p-1-n_{\ast }}(modp).$$ 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 $$\frac{1}{p^{n-l}}\sum _{l<k\le n}\left(\genfrac{}{}{0ex}{}{p^{a}n-d}{p^{a}k-d}\right)(-1{)}^{pk}(\genfrac{}{}{0ex}{}{k-1}{l})=\frac{(-1{)}^{l-1}n!}{l!(n-l)}{B}_{p-n+l}(modp)$$ for all d=1,...,max{p^{a-2},1}.

38 pages

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