Real Solutions of the First Painlevé Equation with Large Initial Data

We consider three special cases of the initial value problem of the first Painlevé (PI) equation. Our approach is based on the method of uniform asymptotics introduced by Bassom et al. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.     

On Airy Solutions of the Second Painlevé Equation

In this paper, we discuss Airy solutions of the second Painlevé equation (P_II) and two related equations, the Painlevé XXXIV equation () and the Jimbo–Miwa–Okamoto σ form of P_II (S_II), are discussed. It is shown that solutions that depend only on the Airy function have a completely different structure to those that involve a linear combination of the Airy functions and . For all three equations, the special solutions that depend only on are tronquée solutions, i.e., they have no poles in a sector of the complex plane. Further, for both and S_II, it is shown that among these tronquée solutions there is a family of solutions that have no poles on the real axis.     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

q‐pseudoconvex and q‐holomorphically convex domains

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q‐pseudoconvexity and q‐holomorphic convexity: we prove that any open subset $\mathrm{\Omega }?{\mathbb{C}}^n$ with smooth boundary, strictly q‐pseudoconvex, is $(q+1)$‐holomorphically convex; moreover, assuming that Ω verifies an additional assumption, we prove that it is q‐holomorphically convex. We also prove that any open subset of ${\mathbb{C}}^n$ is n‐holomorphically convex.     

Monotone Solutions of a Higher-Order Neutral Difference Equation

A real sequence x _k is said to be (*)-monotone with respect to a sequence p _k and a positive integer if x _k > 0 and . This paper is concerned with the existence of (*)-monotone solutions of a neutral difference equation. Existence criteria are derived by means of a comparison theorem and by establishing explicit existence criteria for positive and/or bounded solutions of a majorant recurrence relation.     

q‐Voronovskaya type theorems for q‐Baskakov operators

In the present paper, we prove quantitative q‐Voronovskaya type theorems for q‐Baskakov operators in terms of weighted modulus of continuity. We also present a new form of Voronovskaya theorem, that is, q‐Grüss‐Voronovskaya type theorem for q‐Baskakov operators in quantitative mean. Hence, we describe the rate of convergence and upper bound for the error of approximation, simultaneously. Our results are valid for the subspace of continuous functions although classical ones is valid for differentiable functions. Copyright © 2015 John Wiley & Sons, Ltd.     

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.     

On Non-Negative Measurable Solutions of a Difference Functional Equation

Following M. Laczkovich [5] we consider the iterative functional equation

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

On One-Parameter Families of Painlevé III

Albrecht, Mansfield, and Milne developed a direct method with which one can calculate special integrals of polynomial type (also known as one-parameter family conditions, Darboux polynomials, eigenpolynomials, or algebraic invariant curves) for nonlinear ordinary differential equations of polynomial type. We apply this method to the third Painlevé equation and prove that for the generic case, the set of known one-parameter family conditions is complete.     

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.     

一阶迭代微分方程的解析解

讨论了一阶迭代微分方程解析解的存在性,通过构造一个辅助方程的幂级数解给出该方程的解析解.     

The Painlevé Property and Hirota's Method

The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N-soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self-truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.     

The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability

The characteristic feature of the so-called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly-Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.     

一类高阶有理差分方程解的收敛性(英文)

本文考查下列高阶有理差分方程xn+1=(α+B1xn-1+B3xn-3+…+B2k+1xn-2k-1)/(A+B0xn+B2xn-2+…+B2kxn-2k,),n=0,1,…,其中k是非负整数,参数α,A,Bi,i=0,1,2,…,2k+1和初始值x-2k-1,x-2k,x-2k+1,…,x0是非负实数.给出充分条件,在此条件下当且仅当∑k+1i=1B2i-1=A时,方程的每个解收敛于方程的一个二周期解.     

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

The starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 × 2 isomonodromic Fuchsian systems associated to the Painlevé VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painlevé VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard’s solutions of Painlevé VI.     

一类高阶差分方程周期解的存在性

本文应用临界点理论获得了一类高阶差分方程k∑i=0(xn-i xn i) f(xn 1,xn,xn-1) =0, n∈z,k∈N非平凡M-周期解存在的充分条件.     

Painlevé III Asymptotics of Hankel Determinants for a Perturbed Jacobi Weight

We study the Hankel determinants associated with the weight where , , , is analytic in a domain containing [ ? 1, 1] and for . In this paper, based on the Deift–Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as and . We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo–Miwa–Okamoto σ‐function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.