Showing papers by "Zhi-Wei Sun published in 2003"

Combinatorial identities in dual sequences

Abstract: In this paper we derive a general combinatorial identity in terms of polynomials with dual sequences of coefficients. Moreover, combinatorial identities involving Bernoulli and Euler polynomials are deduced. Also, various other known identities are obtained as particular cases.

Unification of zero-sum problems, subset sums and covers of ℤ

Abstract: In combinatorial number theory, zero-sum problems, subset sums and covers of the integers are three different topics initiated by P. Erdös and investigated by many researchers; they play important roles in both number theory and combinatorics. In this paper we announce some deep connections among these seemingly unrelated fascinating areas, and aim at establishing a unified theory! Our main theorem unifies many results in these three realms and also has applications in many aspects such as finite fields and graph theory. To illustrate this, here we state our extension of the Erdös-Ginzburg-Ziv theorem: If A = {as(mod ns)}s=1 covers some integers exactly 2p − 1 times and others exactly 2p times, where p is a prime, then for any c1, · · · , ck ∈ Z/pZ there exists an I ⊆ {1, · · · , k} such that ∑ s∈I 1/ns = p and ∑ s∈I cs = 0.

Values of Lucas Sequences Modulo Primes

Abstract: Let p be an odd prime, and a, b be two integers. It is the purpose of the paper to determine the values of u (p±1)/2 (a,b) (mod p), where {u n (a,b)} is the Lucas sequence given by u 0 (a,b) = 0, u 1 (a,b) = 1 and u n+1 (a,b) = bu n (a, b) - au n-1 (a,b) (n > 1). In the case a = -c 2 , a reciprocity law is established. As applications we obtain the criteria for p|u (p-1)/4 (a,b) (if p ≡ 1 (mod 4)) and for k E Q 0 (p) and k E Q 1 (p), where Q 0 (p) and Q 1 (p) are defined as in [10].

Zero-sum problems for abelian p-groups and covers of the integers by residue classes

Abstract: Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.

On the Herzog-Sch\"onheim conjecture for uniform covers of groups

Abstract: Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Schonheim conjectured that if $\Cal A=\{a_iG_i\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\max_{1\le j\le k}|\{1\le i\le k:n_i=n_j\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\min_{1\ls i\ls k}\log n_i=O(M\log^2 M)$ where the O-constant is absolute.