scholarly journals Discrete orthogonal polynomials and difference equations of several variables

2007 ◽  
Vol 212 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Plamen Iliev ◽  
Yuan Xu
Keyword(s):  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Several Variables ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal

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.

Difference Equations and Discriminants for Discrete Orthogonal Polynomials

2005 ◽  
Vol 8 (4) ◽  
pp. 475-502 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Inna Nikolova ◽  
Plamen Simeonov
Keyword(s):  
Orthogonal Polynomials ◽  
Difference Equations ◽  
Discrete Orthogonal Polynomials ◽  
Discrete Orthogonal
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