**CRM Monograph Series**

Volume: 28;
2009;
127 pp;
Hardcover

MSC: Primary 33; 11; 26; 15;
**Print ISBN: 978-0-8218-4878-4
Product Code: CRMM/28**

List Price: $54.00

Individual Member Price: $43.20

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# Skew-Orthogonal Polynomials and Random Matrix Theory

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*Saugata Ghosh*

A co-publication of the AMS and Centre de Recherches Mathématiques

Orthogonal polynomials satisfy a three-term recursion relation
irrespective of the weight function with respect to which they are
defined. This gives a simple formula for the kernel function, known in
the literature as the Christoffel–Darboux sum. The availability
of asymptotic results of orthogonal polynomials and the simple
structure of the Christoffel–Darboux sum make the study of
unitary ensembles of random matrices relatively straightforward.

In this book, the author develops the theory of skew-orthogonal
polynomials and obtains recursion relations which, unlike orthogonal
polynomials, depend on weight functions. After deriving reduced
expressions, called the generalized Christoffel–Darboux formulas
(GCD), he obtains universal correlation functions and non-universal
level densities for a wide class of random matrix ensembles using the
GCD.

The author also shows that once questions about higher order
effects are considered (questions that are relevant in different
branches of physics and mathematics) the use of the GCD promises to be
efficient.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Table of Contents

# Table of Contents

## Skew-Orthogonal Polynomials and Random Matrix Theory

#### Readership

Research mathematicians interested in random matrix theory.