scholarly journals Special polynomials associated with the fourth order analogue to the Painlevé equations

2007 ◽  
Vol 363 (5-6) ◽  
pp. 346-355 ◽  
Author(s):  
Nikolai A. Kudryashov ◽  
Maria V. Demina
Keyword(s):  
Fourth Order ◽  
Painlevé Equations ◽  
Painleve Equations ◽  
Special Polynomials

scholarly journals Remarks on the Yablonskii-Vorob’ev polynomials

2000 ◽  
Vol 159 ◽  
pp. 87-111 ◽  
Author(s):  
Makoto Taneda
Keyword(s):  
Algebraic Approach ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Painleve Equation ◽  
Second Painlevé Equation

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

scholarly journals Special polynomials and the Hirota Bilinear relations of the second and the fourth Painlevé equations

2000 ◽  
Vol 159 ◽  
pp. 179-200 ◽  
Author(s):  
Satoshi Fukutani ◽  
Kazuo Okamoto ◽  
Hiroshi Umemura
Keyword(s):  
Recurrence Relation ◽  
Rational Functions ◽  
Painlevé Equations ◽  
Algebraic Proof ◽  
Painleve Equations ◽  
Special Polynomials ◽  
Bilinear Relations ◽  
Hirota Bilinear

We give a purely algebraic proof that the rational functions Pn(t), Qn(t) inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the τ-functions.

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

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.

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

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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