A study on the fourth q-Painlev\'e equation
TLDR
In this article, a q-difference analogue of the fourth Painleve equation is proposed and its symmetry structure and some particular solutions are investigated; see Section 2.1.Abstract:
A q-difference analogue of the fourth Painleve equation is proposed. Its symmetry structure and some particular solutions are investigated.read more
Citations
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Journal ArticleDOI
Geometric aspects of Painlevé equations
Book ChapterDOI
Discrete Painlevé Equations
TL;DR: The Cargese summer school celebrated the 100th anniversary of the Painleve property, the property that was introduced byPainleve and subsequently by Gambier and their school to classify ordinary differential equations (ODEs) according to the singularity behavior of their solutions as mentioned in this paper.
Journal ArticleDOI
Geometric Aspects of Painlev\'e Equations
TL;DR: In this paper, a comprehensive review is given on the current status of achievements in the geometric aspects of the Painleve equations, with a particular emphasis on the discrete painleve equation, which is controlled by the geometry of certain rational surfaces called the spaces of initial values.
Journal ArticleDOI
On a q-Difference Painlevé III Equation: I. Derivation, Symmetry and Riccati Type Solutions
Kenji Kajiwara,Kinji Kimura +1 more
TL;DR: In this paper, a q-difference analogue of the Painleve III equation is considered and its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.
References
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Book
Basic Hypergeometric Series
George Gasper,Mizan Rahman +1 more
TL;DR: In this article, the Askey-Wilson q-beta integral and some associated formulas were used to generate bilinear generating functions for basic orthogonal polynomials.
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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
Roelof Koekoek,René F. Swarttouw +1 more
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
Journal ArticleDOI
Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II
Michio Jimbo,Tetsuji Miwa +1 more
TL;DR: In this article, a series of τ functions parametrized by integers are introduced and their ratios to the original τ function are then shown to be explicit rational expressions in terms of the coefficients of A(x).
Journal ArticleDOI
From soliton equations to integrable cellular automata through a limiting procedure.
TL;DR: A direct connection between a cellular automaton and integrable nonlinear wave equations is shown and a general method for constructing suchintegrable cellular automata and their $N$-soliton solutions is proposed.
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Higher order Painlevé equations of type $A^{(1)}_l$
Masatoshi Noumi,Yasuhiko Yamada +1 more
TL;DR: In this article, a series of nonlinear equations with affine Weyl group symmetry of type $A^{(1)}_l$ was studied, which gave a generalization of Painleve equations to higher orders.