**Contemporary Mathematics**

Volume: 651;
2015;
194 pp;
Softcover

MSC: Primary 34; 37; 39; 35; 14; 17; 33; 41; 81;

Print ISBN: 978-1-4704-1654-6

Product Code: CONM/651

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

**Electronic ISBN: 978-1-4704-2779-5
Product Code: CONM/651.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

# Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations

Share this page *Edited by *
*Anton Dzhamay; Kenichi Maruno; Christopher M. Ormerod*

This volume contains the proceedings of the AMS Special Session on
Algebraic and Analytic Aspects of Integrable Systems and Painlevé
Equations, held on January 18, 2014, at the Joint Mathematics Meetings
in Baltimore, MD.

The theory of integrable systems has been at the forefront of some
of the most important developments in mathematical physics in the last
50 years. The techniques to study such systems have solid foundations
in algebraic geometry, differential geometry, and group representation
theory.

Many important special solutions of continuous and discrete
integrable systems can be written in terms of special functions such
as hypergeometric and basic hypergeometric functions. The analytic
tools developed to study integrable systems have numerous applications
in random matrix theory, statistical mechanics and quantum
gravity. One of the most exciting recent developments has been the
emergence of good and interesting discrete and quantum analogues of
classical integrable differential equations, such as the Painlevé
equations and soliton equations. Many algebraic and analytic ideas
developed in the continuous case generalize in a beautifully natural
manner to discrete integrable systems. The editors have sought to
bring together a collection of expository and research articles that
represent a good cross section of ideas and methods in these active
areas of research within integrable systems and their applications.

#### Readership

Graduate students and research mathematicians interested in integrable systems.

# Table of Contents

## Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Padé Interpolation and Hypergeometric Series 114 free
- 1. Introduction 114
- 2. Padé interpolation problems and their determinant solutions 215
- 3. Hypergeometric series arising from determinants 518
- 4. Three types of very well-poised hypergeometric series 1023
- Acknowledgment 1629
- Appendix A. Condensation of determinants 1629
- Appendix B. Variations of Krattenthaler’s determinant formula 1932
- References 2235

- A 𝑞-analogue of the Drinfeld-Sokolov Hierarchy of Type 𝐴 and 𝑞-Painlevé System 2538
- Fractional Calculus of Quantum Painlevé Systems of Type 𝐴_{𝑙}⁽¹⁾ 3952
- Spectral Curves and Discrete Painlevé Equations 6578
- Geometric Analysis of Reductions from Schlesinger Transformations to Difference Painlevé Equations 87100
- Beta Ensembles, Quantum Painlevé Equations and Isomonodromy Systems 125138
- 1. Introduction 125138
- 2. Beta ensembles, Virasoro constraints and quantum integrable models 128141
- 3. BPZ differential equations for 𝛽-ensembles and CFT [78, 79, 10, 4, 80] 129142
- 4. Results and conjectures for general 𝛽 133146
- 5. Special cases 𝛽=2(𝑐=1) and 𝑐→∞ (𝛽→0 or 𝛽→∞) 142155
- 6. Multidimensional FP operators 146159
- 7. Concluding remarks 148161
- References 149162

- Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation with a One-Sided Non-Zero Boundary Condition 157170
- Back Cover Back Cover1210