p-Catalan numbers and squarefree binomial coefficients

In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p^q divides F(p^q,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ₁(p^q) sufficiently large (τ₁(p^q) computable). Moreover, using the results of the first part, we find n<τ₁(p^q) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10.     

The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.     

On zero varieties of holomorphic functions in Hardy spaces

In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B_n in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈H^p(B_n), 0<p<∞, with . Here N_g(ζ,1) denotes the integrated zero counting function associated with the slice function g_ζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces H^p(B_n), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.     

Note on a Fermat-type diophantine equation

Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1,(k,p)=1.Let C_p⁽ⁱ⁾ be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,ω being the Teichmuller character.In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n<p−4, assume:

(a)

At least one of C_p⁽³⁾,C_p⁽⁵⁾,…,C_p⁽ⁿ⁾ is non-trivial.

(b)

C_p⁽ⁱ⁾=0 for p−n−1?i?p−2.

(c)

for 1?i?n+1.

Let q be an odd prime such that , and such that there is a prime ideal Q over q in whose ideal class is of the form I^pJ where J is non-trivial, not a pth power and J∈C_p⁽³⁾⊕C_p⁽⁵⁾⊕?⊕C_p⁽ⁿ⁾.For such p and q, if x^p+y^p=pcz^p has a non-trivial solution , with (x,y,z)=1, then .Let t(n)=n²²⁴ⁿ⁴. If , then applying a result of Soulé (J. Reine Angew. Math. 517 (1999) 209), we show that the above result holds with only condition (a) because the others are automatically satisfied.We also make a remark about the effect of Soulé's result on the p-divisibility of h_p⁺ (the class number of the maximal real subgroup of ) which is relevant to the existence of integral solutions to x^p+y^p=pcz^p.     

A note on coloring sparse random graphs

Coja-Oghlan and Taraz [Amin Coja-Oghlan, Anusch Taraz, Exact and approximative algorithms for coloring , Random Structures and Algorithms 24 (3) (2004) 259-278] presented a graph coloring algorithm that has expected linear running time for random graphs with edge probability p satisfying np≤1.01. In this work, we develop their analysis by exploiting generating function techniques. We show that, in fact, their algorithm colors G_n,p with the minimal number of colors and has expected linear running time, provided that np≤1.33.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying n_k+1/n_k?λ>1 for all k∈N, belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that lim_r→1(1-r²)^α‖Rf_r‖_p=0 if and only if . These results confirm two conjectures from the following recent paper [S. Stevi?, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].     

Vertex-transitive self-complementary uniform hypergraphs of prime order

For an integer n and a prime p, let . In this paper, we present a construction for vertex-transitive self-complementary k-uniform hypergraphs of order n for each integer n such that for every prime p, where ?=max{k₍₂₎,(k−1)₍₂₎}, and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank k=2^? or k=2^?+1 due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary k-hypergraphs which have prime order p in the case where k=2^? or k=2^?+1 and , and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order , up to isomorphism.     

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, , exist if and only if n is odd with n≠15 and n≠p^α for p an odd prime and α≥2 or with n≠2p^α for p an odd prime and α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K_n−I)^∗, exist if and only if .     

Systems of orthogonal polynomials arising from the modular j-function

Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {P_n,ψ(x)}_n=0^∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)P_n,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.     

Upper bounds on n-dimensional Kloosterman sums

Let p^m be any prime power and K_n(a,p^m) be the Kloosterman sum , where the x_i are restricted to values not divisible by p. Let m,n be positive integers with m?2 and suppose that p^γ||(n+1). We obtain the upper bound , for odd p. For p=2 we obtain the same bound, with an extra factor of 2 inserted.     

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph with m≥n−1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph . The orientable genus problem for with m≥n−1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of when n is even and m≥n, the genus of when n=2^p+2 for p≥3 and m≥n−1, and the genus of when n=2^p+1 for p≥3 and m≥n+1. In all of these cases the genus is the same as the genus of K_m,n, namely ⌈(m−2)(n−2)/4⌉.