Fleck quotients and Bernoulli numbers

Original · EN

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)nk(-1)ᵏ∈. 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=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*+1Bₚ₋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 1pⁿ⁻ˡ∑ₗ<ₖ≤ ₙ pᵃ n-dpᵃ k-d(-1)pkk-1l =(-1)ˡ⁻¹n!l!(n-l)Bₚ₋ₙ₊ₗ (mod p) for all d=1,...,maxpᵃ⁻²,1.