scholarly journals Some Properties of Solutions for Someq-Difference Equations Containing Painlevé Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Peng Jun Zhao ◽  
Hong Yan Xu
Keyword(s):  
Difference Equations ◽  
Painlevé Equation ◽  
Meromorphic Solutions ◽  
Exponent Of Convergence ◽  
Painleve Equation

The existence and growth of meromorphic solutionsf(z)for someq-difference equations are studied, and some estimates for the exponent of convergence of poles ofΔqf,Δq2f,Δqf/f, andΔq2f/fare also obtained. Our theorems are improvements and extensions of the previous results.

Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation

2020 ◽  
pp. 2150029 ◽  
Author(s):  
Jie hu ◽  
Galina Filipuk ◽  
Yang Chen
Keyword(s):  
Difference Equation ◽  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Painlevé Equation ◽  
Bäcklund Transformations ◽  
Recurrence Coefficients ◽  
Backlund Transformations ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal ◽  
Painleve Equation

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the [Formula: see text]-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.

scholarly journals ON GENERAL REPRESENTATION OF THE MEROMORPHIC SOLUTIONS OF HIGHER ANALOGUES OF THE SECOND PAINLEVE EQUATION

1997 ◽  
Vol 2 (1) ◽  
pp. 61-65
Author(s):  
V. I. Gromak ◽  
T. S. Stepanova
Keyword(s):  
Mathematical Modelling ◽  
Painlevé Equation ◽  
Meromorphic Solutions ◽  
General Representation ◽  
Modelling Analysis ◽  
Painleve Equation ◽  
Second Painlevé Equation

„On general representation of the meromorphic solutions of higher analogues of the second painleve equation" Mathematical Modelling Analysis, 2(1), p.61-65

scholarly journals The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng
Keyword(s):  
Fixed Points ◽  
Difference Equations ◽  
Painlevé Equations ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Zeros And Poles

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

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