A study of the associated linear problem for q-P_V

2009/11/30 by Christopher M. Ormerod, Ormerod, Christopher M. · 1 citation
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #math.CA #math.DS

paper · pdf · doi:10.48550/arxiv.0911.5552

28 pages

arxiv created 2009/11/30 · openalex publication_date 2009/11/30 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the associated linear problem for a q-analogue of the fifth Painleve equation (qPV). We identify a lattice of connection preserving deformations in the space of the connection data for the linear problem with the lattice of translational Backlund transformations for qPV, hence, show all translational Backlund transformations possess a Lax pair. We shall show that the big q-Laguerre polynomials, and a suitable generalization, solve a special case of the linear problem, hence, find solutions to qPV in terms of determinants of Hankel matrices with entries consisting of rational or hypergeometric functions.

Citations

• Orthogonal polynomials and their derivatives, I
• A differential equation for orthogonal polynomials
• Ladder Operators for q-orthogonal Polynomials
• Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II

Cited by

• The Lattice Structure of Connection Preserving Deformations for q-Painlevé Equations I

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• On the Rational Solutions of q-Painlevé V Equation
• Discrete Equations from BäckLund Transformations of the Fifth Painlevé Equation
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• A q-analogue of the matrix fifth Painlevé system
• Bäcklund transformations for certain rational solutions of Painlevé VI