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## Details

Genre/Form: | Electronic books Conference papers and proceedings Congresses |
---|---|

Additional Physical Format: | Print version: (OCoLC)978289837 |

Material Type: | Conference publication, Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
D Levi; Pavel Winternitz; Raphaël Rebelo |

ISBN: | 9783319566665 3319566660 |

OCLC Number: | 992780283 |

Notes: | Includes index. |

Description: | 1 online resource. |

Contents: | Preface; Contents; Contributors; Continuous, Discrete and Ultradiscrete Painlevé Equations; 1 Introduction; 2 Painlevé Equations; 2.1 Introduction; 2.2 Special Solutions; 2.2.1 Rational Solutions; 2.2.2 Hypergeometric Solutions; 2.3 Bäcklund Transformations; 2.3.1 A Bäcklund Transformation; 2.3.2 Higher Solutions; 2.3.3 Affine Weyl Group Symmetry; 3 Discrete Painlevé Equations; 3.1 A Brief Introduction with an Example; 3.2 A Discrete Painlevé Equation Arising from the Bäcklund Transformations of the PIV Equation; 3.2.1 Weyl Group; 3.2.2 The Parameter Space of the PIV Equation. 3.2.3 An Additive Difference Painlevé Equation3.3 Classification and Some Properties of the Discrete Painlevé Equations; 3.3.1 Additive Difference; 3.3.2 Multiplicative Difference; 3.3.3 Elliptic Difference; 3.4 A q-Discrete Analogue of the Fourth Painlevé Equation; 3.4.1 Special Solutions of q-Painlevé Equations; 3.5 q-Hypergeometric Type Special Solutions of q-PIV; 4 Ultradiscrete Painlevé Equations; 4.1 Introduction; 4.2 Ultradiscretization; 4.3 Ultradiscrete KdV Equation; 4.4 Ultradiscrete Painlevé Equations (ud-Ps); 4.5 Degeneration Diagram of Ud-Painlevé Equations. 4.6 Special Solutions to the Ud-Painlevé Equations4.7 Ultradiscrete Limit with Sign Variables; 4.8 Answers to the Exercises in Sect. 4; 4.8.1 Solution to Exercise 4.1; 4.8.2 Solution to Exercise 4.3; 4.8.3 Solution to Exercise 4.4; 4.9 Conclusion of Sect. 4; References; Elliptic Hypergeometric Functions; 1 Introduction; 2 Hypergeometric Series; 3 Basic Hypergeometric Series; 4 Elliptic Functions; 5 Elliptic Hypergeometric Series; 6 6j-Symbols and Spiridonov-Zhedanov Biorthogonal Functions; 7 Elliptic Beta Integral; 8 Continuous Biorthogonality; References. Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries1 Introduction: The Meaning of Integrability; 2 Algebraic Entropy; 2.1 Definition and Basic Properties; 2.2 Computational Tools; 2.3 Examples; 3 Generalized Symmetries; 3.1 Generalized Symmetries and Integrability; 3.2 Generalized Symmetries for Differential-Difference Equations; 3.3 Three-Point Generalized Symmetries for Quad Equation; 3.4 Master Symmetries; 3.5 Some Examples; 4 Exercises; 4.1 Algebraic Entropy; 4.2 Generalized Symmetries; References. Introduction to Linear and Nonlinear Integrable Theories in Discrete Complex Analysis1 Introduction; 2 Linear Theory of Discrete Holomorphic Functions; 2.1 Definition of Discrete Harmonic and Discrete Holomorphic Functions; 2.2 3D-Consistency and Integrability; 2.3 Some Further Topics; 3 Discrete Holomorphic Functions and Cross-Ratio Systems; 3.1 Definition of Discrete Conformal Functions; 3.2 3D-Consistency and Bäcklund Transformations; 3.3 Linearization; 3.4 Some Further Topics; 4 Circle Patterns as Discrete Holomorphic Functions; 4.1 Definition of Discrete Conformal Functions. |

Series Title: | CRM series in mathematical physics. |

Responsibility: | Decio Levi, Raphaël Rebelo, Pavel Winternitz, editors. |

### Abstract:

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations.
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