# Differential Algebra, Complex Analysis and Orthogonal Polynomials

Share this page *Edited by *
*Primitivo B. Acosta-Humánez; Francisco Marcellán*

This volume represents the 2007–2008 Jairo Charris Seminar in
Algebra and Analysis on Differential Algebra, Complex Analysis and
Orthogonal Polynomials, which was held at the Universidad Sergio
Arboleda in Bogotá, Colombia.

It provides the state of the art in the theory of Integrable
Dynamical Systems based on such approaches as Differential Galois
Theory and Lie Groups as well as some recent developments in the
theory of multivariable and \(q\)-orthogonal polynomials, weak
Hilbert's 16th Problem, Singularity Theory, Tournaments in flag
manifolds, and spaces of bounded analytic functions on the unit
circle.

The reader will also find survey presentations, an account of
recent developments, and the exposition of new trends in the areas of
Differential Galois Theory, Integrable Dynamical Systems, Orthogonal
Polynomials and Special Functions, and Bloch–Bergman classes of
analytic functions from a theoretical and an applied perspective.

The contributions present new results and methods, as well as
applications and open problems, to foster interest in research in
these areas.

This book is published in cooperation with Instituto de Matemáticas y sus Aplicaciones (IMA)

#### Readership

Graduate students and research mathematicians interested in orthogonal polynomials, differential algebra, and integrability of dynamical systems.

# Table of Contents

## Differential Algebra, Complex Analysis and Orthogonal Polynomials

- Contents v6 free
- Preface vii8 free
- Differential Galois Theory of Algebraic Lie-Vessiot Systems 110 free
- 1. Introduction 211
- 2. Algebraic Groups and Homogeneous Spaces 413
- 3. Differential Algebraic Geometry 817
- 4. Galois theory of Algebraic Lie-Vessiot Systems 1726
- 5. Algebraic Reduction and Integration 3544
- 6. Integrability of Linear Equations 4857
- Appendix A. Stalk formula for affine morphisms 5564
- References 5665

- Recent Trends on Two Variable Orthogonal Polynomials 5968
- 1. Introduction 6069
- 2. Algebraic properties of orthogonal polynomials in two variables 6170
- 3. Orthogonal polynomials in two variables and eigenfunctions of second order partial differential equations 6776
- 4. Extended definition of classical orthogonal polynomials in two variables 7281
- 5. Semiclassical orthogonal polynomials in two variables 7685
- 6. Sobolev orthogonal polynomials in several variables 7887
- 7. Open problems 8392
- 8. Acknowledgements 8493
- References 8493

- On the Integrability of the Riccati Equation 8796
- Two Discrete Systems of q-orthogonal Polynomials 95104
- Like-hyperbolic Bloch-Bergman Classes 103112
- Some words about the application of Tchebycheff systems to Weak Hilbert's 16th Problem 119128
- From the index of a differential operator to the Milnor number of a singularity 129138
- Integrability of dynamical systems through differential Galois theory: A practical guide 143152
- 1. Introduction 144153
- 2. General Non-integrability Theorems 146155
- 3. Homogeneous Potentials and Related Problems 157166
- 4. Hamiltonian Rigid Body Problem 189198
- 5. Cosmological Models 192201
- 6. An Application to Painlevé's Transcendents 195204
- Appendix A. Algorithmic Considerations 201210
- Appendix B. Hypergeometric Equation 211220
- Appendix C. Lamé Equation 212221
- References 215224

- Tournaments and parabolic almost complex structures on flag manifolds 221230