The values at x=0 are called Bernoulli and Euler numbers of order w; when w=1, the polynomials or numbers are called ordinary. When x=0 or w=1, we often suppress that part of the notation; e.g., B (w) n denotes B n (0), En(x) denotes E (1) n (x), and Bn denotes B (1) n (0). These numbers have been extensively studied and many congruences for them are known. Among the most important results are ...

and Applied Analysis 3 Remark 5. Assumption (13) holds if the system has a unique solution. For brevity, we rewrite system (2)∼(5) in the vector form by introducing W(x, t) = (u(x, t), n(x, t)) and Y(x, t) = (a(x, t), b(x, t)). We denote by E 0 = H 2 ⋂H 1 0 (Ω) × H 1 0 (Ω) the space of vector functionsW(x, t) = (u(x, t), n(x, t)) with norm E0 = {‖u‖ 2 H 2 + ‖n‖ 2 H 1} 1/2 . (14) Similarly, we d...

Abstract. In this paper, we consider the regularity of weak solutions u ∈ W (R ) ∩ W (R ) of the elliptic partial differential equation −∆pu−∆qu = f(x), x ∈ R , where 1 < q < p < N . We prove that these solutions are locally in C and decay exponentially at infinity. Furthermore, we prove the regularity for the solutions u ∈ W (R ) ∩W (R ) of the following equations −∆pu−∆qu = f(x, u), x ∈ R , w...

1. INTRODUCTION Horadam [1] has generalized two theorems of Subba Rao [3] which deal with some asymptotic p r o p e r t i e s of Fibonacci numbers. Horadam defined a sequence {w (n 2) where a , a are the roots of x 2-P 21 x + P 2 2 = 0. We shall let 06 """ LX r\ r\ UO r\-| • Horadam established two theorems for {w n }: I. The number of terms of {w n } not exceeding N is asymptotic to log(Nd/(P ...

In this paper, we consider the orthogonal polynomials with respect to weight w(x)=w(x;s):=xλe−N[x+s(x3−x)],x∈R+, where λ&gt;0, N&gt;0 and 0≤s≤1. By using ladder operator approach, obtain a pair of second-order nonlinear difference equations differential–difference satisfied by recurrence coefficients αn(s) βn(s). We also establish relation between associated Hankel determinant coefficients. Fro...

Definition 1.3. Suppose Z ∈ 2ω. A test relative to Z, or Z-test, is a set W ⊆ ω × 2<ω which is recursively enumerable in Z. For X ∈ 2ω, X passes a test W if and only if there is an n such that X 6∈ [Wn]. Definition 1.4. Suppose that m represents the measure μ on 2ω and that W is an m-test. • W is correct for μ if and only if for all n, ∑ σ∈Wn μ([σ]) ≤ 2 −n. • W is Solovay-correct for μ if and o...

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Laplacians (−Δ)Ωs, with Ω⊂RN. Recall that are generated by Lévy-type processes which not allowed jump outside Ω. consider weak equation (−Δ)Ωsw(x)=p.v.∫Ωw(x)−w(y)|x−y|N+2sdy=f(x), s∈(0,1) and Ω⊂RN, subject zero Neumann or Dirichlet conditions. The conditions defined considering w as we...

x(n + 1) : x(n)exp Jr (n ) (1 x(n) K(n) ] J ' where {r(n)} and {K(n)} are positive w-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable w-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect. (~) 2003 Elsevier Science Ltd. All rights reserved.

For N ≥ 3 and p > 1, we consider the nonlinear Schrödinger equation i∂tw +∆xw + V (x) |w|p−1 w = 0 where w = w(t, x) : R× R → C with a potential V that decays at infinity like |x|−b for some b ∈ (0, 2). A standing wave is a solution of the form w(t, x) = eu(x) where λ > 0 and u : R → R. For 1 < p < 1 + (4− 2b)/(N − 2), we establish the existence of a C1-branch of standing waves parametrized by ...