The continuous Painlevé equations and the Garnier system

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

Open Mathematics ◽

10.1515/math-2019-0079 ◽

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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BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

Annales Henri Poincaré ◽

10.1007/s00023-021-01034-3 ◽

2021 ◽

Author(s):

Giulio Bonelli ◽

Fabrizio Del Monte ◽

Alessandro Tanzini

Keyword(s):

Topological String ◽

Cluster Algebra ◽

Painlevé Equations ◽

Partition Functions ◽

Quantum Field Theories ◽

Particle Spectrum ◽

Yang Mills ◽

Painleve Equations ◽

Rank One ◽

Bilinear Equations

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.

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$q$ -Racah Ensemble and Discrete Painlevé Equation

International Mathematics Research Notices ◽

10.1093/imrn/rnz211 ◽

2019 ◽

Vol 2020 (24) ◽

pp. 9797-9843 ◽

Cited By ~ 1

Author(s):

Anton Dzhamay ◽

Alisa Knizel

Keyword(s):

Orthogonal Polynomial ◽

Lax Pair ◽

Painlevé Equations ◽

Painlevé Equation ◽

Painleve Equations ◽

Symmetry Structure ◽

The One ◽

The Relationship ◽

Gap Probabilities ◽

Discrete Painlevé Equation

Abstract The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

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