Discrete Orthogonal Polynomials. (AM-164): Asymptotics and Applications (AM-164) (Annals of Mathematics Studies, 164) by J. Baik (PDF)

Description

This book describes the theory and applications of discrete orthogonal polynomials–polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

User’s Reviews

Editorial Reviews: Review “This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications.”—H.J. Glaeske, Zentralblatt MATH”The book includes the authors’ own research results developed over the last years. Their approach and proofs are straightforward constructive making this monograph accessible and valuable to undergraduate and graduate students, PhD students and researchers involved in the asymptotic analysis of systems of discrete orthogonal polynomials.”—Octavian Agratini, Mathematica”This book is very interesting. It pulls together several methods developed during the last twenty years and makes them into a general theory. Thanks to this general theory, several open problems are solved. The organization of the book is very nice.”—Sylvie Corteel, Mathematical Reviews About the Author J. Baik is Associate Professor of Mathematics at the University of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universität Bochum in Bochum, Germany. K. T.-R. McLaughlin is Professor of Mathematics at the University of Arizona. P. D. Miller is Associate Professor of Mathematics at the University of Michigan.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book deals mostly with special cases of Krawtchouk and Hahnpolynomials that have been developed fairly recently.Even with Mathematica to program the forms in this book are very difficult: the recurrences are nearly impossible to program as presented.The classical orthogonal polynomials are in no way bridged so that a graduate student would have a chance with this book.The Hahn hexagonal random tiling is interesting, but the form presented is Gaussian based. The combinatorial arguments aren’t presented in a way thatcan be easily understand or applied.So this makes a good library book, but individuals should probably not buy it.

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