**Contemporary Mathematics**

Volume: 578;
2012;
254 pp;
Softcover

MSC: Primary 30; 33; 42; 41;

Print ISBN: 978-0-8218-6896-6

Product Code: CONM/578

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-0-8218-9102-5
Product Code: CONM/578.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

# Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications

Share this page *Edited by *
*J. Arvesú; G. López Lagomasino*

This volume contains the proceedings of the 11th International
Symposium on Orthogonal Polynomials, Special Functions, and their
Applications, held August 29–September 2, 2011, at the Universidad
Carlos III de Madrid in Leganés, Spain.

The papers cover asymptotic properties of polynomials on curves of
the complex plane, universality behavior of sequences of orthogonal
polynomials for large classes of measures and its application in
random matrix theory, the Riemann–Hilbert approach in the study of
Padé approximation and asymptotics of orthogonal polynomials,
quantum walks and CMV matrices, spectral modifications of linear
functionals and their effect on the associated orthogonal polynomials,
bivariate orthogonal polynomials, and optimal Riesz and logarithmic
energy distribution of points. The methods used include potential
theory, boundary values of analytic functions, Riemann–Hilbert
analysis, and the steepest descent method.

#### Readership

Graduate students and research mathematicians interested in special functions and orthogonal polynomials.

# Table of Contents

## Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications

- Preface ix10 free
- Life and work (so far) of Paco Marcellán 112 free
- Life of Paco Marcellán 112
- 1. The early years 112
- 2. At the university 213
- 3. Doctoral dissertation 314
- 4. Research career 516
- 5. Professional career 617
- 6. Non-academic activities 617
- Work of Paco Marcellán 819
- 7. Cassinian curves and lemniscates 819
- 8. Bar-le-Duc: the start of OPSFA 920
- 9. Perturbations of orthogonal polynomials 1021
- 10. Sobolev orthogonal polynomials 1122
- 11. Other topics 1425
- 12. Paco as an organizer 1526
- 13. Epilog 1627
- References 1627

- Asymptotics of 𝐿_{𝑝}-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states 1930
- The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere 3142
- Spectral transformations of hermitian linear functionals 6374
- Numerical study of higher order analogues of the Tracy–Widom distribution 8394
- Comb functions 99110
- 1. Introduction 99110
- 2. Comb representation of 𝐿𝑃 entire functions 102113
- 3. MacLane’s theorem 104115
- 4. Representation of Green’s and Martin’s functions 108119
- 5. More general combs 112123
- 6. Uniform approximation and extremal problems 113124
- 7. Spectral theory and harmonic analysis 115126
- References 116127

- Orthogonality relations for bivariate Bernstein-Szegő measures 119130
- Quantum walks and CMV matrices 133144
- 1. Introduction and contents of the paper 133144
- 2. Szegő polynomials and CMV matrices 134145
- 3. Traditional quantum walks 136147
- 4. Quantum walks resulting from a probability measure 137148
- 5. The Verblunsky coefficients for Riesz’s measure 138149
- 6. Building the backbone 138149
- 7. Building up the chain 139150
- References 140151

- Discrete beta ensembles based on Gauss type quadratures 143154
- Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later 165176
- 1. Padé approximants to algebraic functions 165176
- 2. Statement of the problem 168179
- 3. Heine and Stieltjes, or asymptotic analysis based on the Liuoville-Green approximation 170181
- 4. Case of 𝑝=3: Chebotarev compact and the Riemann surface 177188
- 5. Riemann and Hilbert, or the non-linear steepest descent analysis 181192
- 6. Wrapping up, or matching the asymptotic formulas and the Nuttall conjecture 190201
- References 191202

- Orthogonal Polynomials and 𝑆-curves 195206
- 1. Complex orthogonal polynomials and 𝑆-curves 195206
- 2. Padé approximants for functions with branch points 198209
- 3. An existence theorem for an 𝑆-curve in harmonic external field. 200211
- 4. Critical measures and equilibrium measures of 𝑆-curves 203214
- 5. Rational external fields 206217
- 6. Green’s and the vector equilibrium problems 209220
- 7. Strong asymptotics 216227
- 8. Generalized Jacobi polynomials 220231
- 9. Proof of the existence theorem 223234
- References 236247

- Fast decreasing and orthogonal polynomials 241252
- 1. Fast decreasing polynomials 241252
- 2. Quasi-uniform zero spacing of orthogonal polynomials 243254
- 3. Christoffel functions 244255
- 4. Nonsymmetric fast decreasing polynomials 247258
- 5. Fast decreasing polynomials on the complex plane 247258
- 6. Christoffel functions on a system of Jordan curves 248259
- 7. Universality 251262
- 8. The Levin-Lubinsky fine zero spacing theorem 252263
- References 253264