Corpus ID: 237513877

# Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlev\'e equations via the geometric approach

@inproceedings{Dzhamay2021DifferentialEF,
title={Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlev\'e equations via the geometric approach},
author={Anton Dzhamay and Galina Filipuk and Alexander Stokes},
year={2021}
}
• Published 14 September 2021
• Mathematics, Physics
In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of Okamoto’s space of initial values. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in [HC17], we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with… Expand
1 Citations

#### Figures from this paper

Different Hamiltonians for the Painlev\'e ${\text{P}_{\mathrm{IV}}}$ equation and their identification using a geometric approach
• Physics, Mathematics
• 2021
It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique – there are many very differentExpand

#### References

SHOWING 1-10 OF 45 REFERENCES
Differential and difference equations for recurrence coefficients of orthogonal polynomials with hypergeometric weights and Bäcklund transformations of the sixth Painlevé equation
• Mathematics
• Random Matrices: Theory and Applications
• 2020
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.]Expand
The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant
• Mathematics
• 2017
In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight w(x,s) = w(x; λ,s) := xλe−N(x+s(x2−x)), 0 ≤ x −1, 0 ≤ sExpand
Painlev'e di erential equations in the complex plane
• Mathematics
• 2002
This book is the first comprehensive treatment of Painleve differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, theExpand
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations
• Physics, Mathematics
• Symmetry, Integrability and Geometry: Methods and Applications
• 2018
Although the theory of discrete Painleve (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able toExpand
Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• Mathematics
• 2018
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear differenceExpand
Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations
Abstract: We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonicalExpand
The first, second and fourth Painlev\'{e} equations on weighted projective spaces
The first, second and fourth Painlev\'{e} equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weightsExpand
Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Mathematics, Physics
• 2017
It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel–Roberts–Thompson mappings, can be deautonomized to discrete Painleve equations. However, theExpand
Families of Okamoto–Painlevé pairs and Painlevé equations
In [as reported by Saito et al. (J. Algebraic Geom. 11:311–362, 2002)], generalized Okamoto–Painlevé pairs are introduced as a generalization of Okamoto’s space of initial conditions of PainlevéExpand
From Gauss to Painlevé : a modern theory of special functions : dedicated to Tosihusa Kimura
1. Elements of Differential Equations.- 1.1 Cauchy's existence theorem.- 1.2 Linear equations.- 1.3 Local behavior around regular singularities (Frobenius's method).- 1.4 Fuchsian equations.- 1.5Expand