scholarly journals Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painlevé equations

2021 ◽  
Vol 494 (2) ◽  
pp. 124605 ◽  
Author(s):  
Amílcar Branquinho ◽  
Ana Foulquié Moreno ◽  
Manuel Mañas
Keyword(s):  
Eigenvalue Problems ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Biorthogonal Polynomials ◽  
Matrix Biorthogonal Polynomials

scholarly journals Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

2011 ◽  
Vol 467 (2136) ◽  
pp. 3443-3468 ◽  
Author(s):  
Nalini Joshi ◽  
Yang Shi
Keyword(s):  
Exact Solutions ◽  
Linear Problem ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Group Symmetry ◽  
Rational Solutions ◽  
Painleve Equations ◽  
Discrete Painlevé Equations ◽  
Painleve Equation ◽  
Second Painlevé Equation

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Hierarchies of q-discrete Painlevé equations

2020 ◽  
Vol 27 (3) ◽  
pp. 453-477 ◽  
Author(s):  
Huda Alrashdi ◽  
Nalini Joshi ◽  
Dinh Thi Tran
Keyword(s):  
Painlevé Equations ◽  
Painleve Equations ◽  
Discrete Painlevé Equations
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