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      On Fleck quotients

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          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 \(\Q(\zeta_p)\). In addition, generalized Fleck quotients are also studied in this paper.

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          Most cited references 10

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          Combinatorial identities in dual sequences

           Zhi-Wei Sun (2003)
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            On the sum Σ k≡r(modm) ( k n ) and related congruencesand related congruences

             Zhi-Wei Sun (2002)
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              New identities involving Bernoulli and Euler polynomials

               Hao Pan,  Zhi-Wei Sun (2006)
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                Author and article information

                Journal
                2006-03-20
                2007-04-30
                Article
                10.4064/aa127-4-3
                math/0603462
                Custom metadata
                11B65; 05A10; 11A07; 11B68; 11B73; 11L05; 11S99
                Acta Arith. 127(2007), no.4, 337-363
                28 pages
                math.NT math.CO

                Combinatorics, Number theory

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