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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.

### Most cited references10

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### Combinatorial identities in dual sequences

(2003)
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### On the sum Σ k≡r(modm) ( k n ) and related congruencesand related congruences

(2002)
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• Record: found

### New identities involving Bernoulli and Euler polynomials

(2006)
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### Author and article information

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

Combinatorics, Number theory