On Fleck quotients

Abstract

Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\lfloor(n-1)/(p-1)\rfloor} \sum_{k=r(mod p)}\binom {n}{k}(-1)^k\in Z.$$ We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\le n\le p$ then $$\sum_{k=1}^n(-1)^{pk-1}\binom{pn-1}{pk-1} \equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. We also establish the Kummer-type congruence $F_p(n+p^a(p-1),r)\equiv F_p(n,r) (mod p^a)$ for $a=1,2,3,...$, and reveal some connections between Fleck's quotients and class numbers of the quadratic fields $\Q(\sqrt{\pm p})$ and the $p$-th cyclotomic field $\Q(ζ_p)$. In addition, generalized Fleck quotients are also studied in this paper.