Special Functions and Orthogonal Polynomials

Author: Refaat El Attar

Publisher: Lulu.com

ISBN: 1411666909

Category: Mathematics

Page: 310

View: 5322

(308 Pages). This book is written to provide an easy to follow study on the subject of Special Functions and Orthogonal Polynomials. It is written in such a way that it can be used as a self study text. Basic knowledge of calculus and differential equations is needed. The book is intended to help students in engineering, physics and applied sciences understand various aspects of Special Functions and Orthogonal Polynomials that very often occur in engineering, physics, mathematics and applied sciences. The book is organized in chapters that are in a sense self contained. Chapter 1 deals with series solutions of Differential Equations. Gamma and Beta functions are studied in Chapter 2 together with other functions that are defined by integrals. Legendre Polynomials and Functions are studied in Chapter 3. Chapters 4 and 5 deal with Hermite, Laguerre and other Orthogonal Polynomials. A detailed treatise of Bessel Function in given in Chapter 6.
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Orthogonal Polynomials and Special Functions

Author: Richard Askey

Publisher: SIAM

ISBN: 0898710189

Category: Mathematics

Page: 110

View: 9514

This volume presents the idea that one studies orthogonal polynomials and special functions to use them to solve problems.
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Difference Equations, Special Functions and Orthogonal Polynomials

Proceedings of the International Conference, Munich, Germany, 25-30 July 2005

Author: Saber Elaydi

Publisher: World Scientific

ISBN: 9812706437

Category: Science

Page: 773

View: 8952

This volume contains talks given at a joint meeting of three communities working in the fields of difference equations, special functions and applications (ISDE, OPSFA, and SIDE). The articles reflect the diversity of the topics in the meeting but have difference equations as common thread. Articles cover topics in difference equations, discrete dynamical systems, special functions, orthogonal polynomials, symmetries, and integrable difference equations.
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Special Functions 2000: Current Perspective and Future Directions

Author: Joaquin Bustoz,Mourad E.H. Ismail,Sergei Suslov

Publisher: Springer Science & Business Media

ISBN: 9401008183

Category: Mathematics

Page: 520

View: 4861

The Advanced Study Institute brought together researchers in the main areas of special functions and applications to present recent developments in the theory, review the accomplishments of past decades, and chart directions for future research. Some of the topics covered are orthogonal polynomials and special functions in one and several variables, asymptotic, continued fractions, applications to number theory, combinatorics and mathematical physics, integrable systems, harmonic analysis and quantum groups, Painlevé classification.
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Orthogonal Polynomials and Special Functions

Computation and Applications

Author: European summer school on orthogonal polynomials and special functions,Francisco Marcellàn

Publisher: Springer Science & Business Media

ISBN: 3540310622

Category: Mathematics

Page: 418

View: 1417

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askey’s scheme using Leonard pairs, and non-linear special functions associated with the Painlevé equations.
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Special Functions and Orthogonal Polynomials

Author: Richard Beals,Roderick Wong

Publisher: Cambridge University Press

ISBN: 1316578054

Category: Mathematics

Page: N.A

View: 7531

The subject of special functions is often presented as a collection of disparate results, rarely organized in a coherent way. This book emphasizes general principles that unify and demarcate the subjects of study. The authors' main goals are to provide clear motivation, efficient proofs, and original references for all of the principal results. The book covers standard material, but also much more. It shows how much of the subject can be traced back to two equations - the hypergeometric equation and confluent hypergeometric equation - and it details the ways in which these equations are canonical and special. There is extended coverage of orthogonal polynomials, including connections to approximation theory, continued fractions, and the moment problem, as well as an introduction to new asymptotic methods. There are also chapters on Meijer G-functions and elliptic functions. The final chapter introduces Painlevé transcendents, which have been termed the 'special functions of the twenty-first century'.
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Special Functions and Orthogonal Polynomials

AMS Special Session on Special Functions and Orthogonal Polynomials, April 21-22, 2007, Tucson, Arizona

Author: Diego Dominici,Robert Sullivan Maier

Publisher: American Mathematical Soc.

ISBN: 0821846507

Category: Mathematics

Page: 218

View: 3126

This volume contains fourteen articles that represent the AMS Special Session on Special Functions and Orthogonal Polynomials, held in Tucson, Arizona in April of 2007. It gives an overview of the modern field of special functions with all major subfields represented, including: applications to algebraic geometry, asymptotic analysis, conformal mapping, differential equations, elliptic functions, fractional calculus, hypergeometric and $q$-hypergeometric series, nonlinear waves, number theory, symbolic and numerical evaluation of integrals, and theta functions. A few articles are expository, with extensive bibliographies, but all contain original research. This book is intended for pure and applied mathematicians who are interested in recent developments in the theory of special functions. It covers a wide range of active areas of research and demonstrates the vitality of the field.
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Laredo Lectures on Orthogonal Polynomials and Special Functions

Author: Renato Alvarez-Nodarse,Francisco Marcellán,Walter van Assche

Publisher: Nova Publishers

ISBN: 9781594540097

Category: Mathematics

Page: 210

View: 480

This new book presents research in orthogonal polynomials and special functions. Recent developments in the theory and accomplishments of the last decade are pointed out and directions for research in the future are identified. The topics covered include matrix orthogonal polynomials, spectral theory and special functions, Asymptotics for orthogonal polynomials via Riemann-Hilbert methods, Polynomial wavelets and Koornwinder polynomials.
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Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications

11th International Symposium, August 29-September 2, 2011, Universidad Carlos III de Madrid, Leganés, Spain

Author: Jorge Arvesœ,Guillermo Lopez Lagomasino

Publisher: American Mathematical Soc.

ISBN: 0821868969

Category: Mathematics

Page: 254

View: 5653

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 Leganes, 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 Pade 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.
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Orthogonal Polynomials and Special Functions

Leuven 2002

Author: Erik Koelink,Walter Van Assche

Publisher: Springer

ISBN: 3540449450

Category: Mathematics

Page: 250

View: 4984

The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. Thenbsp;volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring onlynbsp;a basic knowledge of analysis and algebra, and each includes many exercises.
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An Introduction to Orthogonal Polynomials

Author: Theodore S Chihara

Publisher: Courier Corporation

ISBN: 0486141411

Category: Mathematics

Page: 272

View: 8713

Concise introduction covers general elementary theory, including the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula, special functions, and some specific systems. 1978 edition.
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Orthogonal Polynomials of Several Variables

Author: Charles F. Dunkl,Yuan Xu

Publisher: Cambridge University Press

ISBN: 1107071895

Category: Mathematics

Page: 426

View: 2383

Updated throughout, this revised edition contains 25% new material covering progress made in the field over the past decade.
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Coimbra Lecture Notes on Orthogonal Polynomials

Author: Amilcar Jose Pinto Lopes Branquinho

Publisher: Nova Publishers

ISBN: 9781600219726

Category: Science

Page: 233

View: 5818

Orthogonal Polynomials and Special Functions (OPSF) have a very rich history, going back to 19th century when mathematicians and physicists tried to solve the most important deferential equations of mathematical physics. Hermite-Padé approximation was also introduced at that time, to prove the transcendence of the remarkable constant e (the basis of the natural logarithm). Since then OPSF has developed to a standard subject within mathematics, which is driven by applications. The applications are numerous, both within mathematics (e.g. statistics, combinatory, harmonic analysis, number theory) and other sciences, such as physics, biology, computer science, chemistry. The main reason for the fact that OPSF has been so successful over the centuries is its usefulness in other branches of mathematics and physics, as well as other sciences. There are many different aspects of OPSF. Some of the most important developments for OPSF are related to the theory of rational approximation of analytic functions, in particular the extension to simultaneous rational approximation to a system of functions. Important tools for rational approximation are Riemann-Hilbert problems, the theory of orthogonal polynomials, logarithmic potential theory, and operator theory for difference operators. This new book presents the latest research in the field.
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Orthogonal Polynomials and Painlevé Equations

Author: Walter Van Assche

Publisher: Cambridge University Press

ISBN: 1108441947

Category: Mathematics

Page: N.A

View: 5818

There are a number of intriguing connections between Painlev� equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlev� equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlev� transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlev� equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlev� equations.
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Symmetric Functions and Orthogonal Polynomials

Author: Ian Grant Macdonald

Publisher: American Mathematical Soc.

ISBN: 9780821882719

Category: Mathematics

Page: 53

View: 7224

One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials, has long been known to be connected to combinatorics, representation theory and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.
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Inzell Lectures on Orthogonal Polynomials

Author: Wolfgang zu Castell,Frank Filbir,Brigitte Forster

Publisher: Nova Publishers

ISBN: 9781594541087

Category: Mathematics

Page: 199

View: 7107

Based on the success of Fourier analysis and Hilbert space theory, orthogonal expansions undoubtedly count as fundamental concepts of mathematical analysis. Along with the need for highly involved functions systems having special properties and analysis on more complicated domains, harmonic analysis has steadily increased its importance in modern mathematical analysis. Deep connections between harmonic analysis and the theory of special functions have been discovered comparatively late, but since then have been exploited in many directions. The Inzell Lectures focus on the interrelation between orthogonal polynomials and harmonic analysis.
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Orthogonal Polynomials

Theory and Practice

Author: Paul Nevai

Publisher: Springer Science & Business Media

ISBN: 9400905017

Category: Mathematics

Page: 488

View: 1408

This volume contains the Proceedings of the NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" held at The Ohio State University in Columbus, Ohio, U.S.A. between May 22,1989 and June 3,1989. The Advanced Study Institute primarily concentrated on those aspects of the theory and practice of orthogonal polynomials which surfaced in the past decade when the theory of orthogonal polynomials started to experience an unparalleled growth. This progress started with Richard Askey's Regional Confer ence Lectures on "Orthogonal Polynomials and Special Functions" in 1975, and subsequent discoveries led to a substantial revaluation of one's perceptions as to the nature of orthogonal polynomials and their applicability. The recent popularity of orthogonal polynomials is only partially due to Louis de Branges's solution of the Bieberbach conjecture which uses an inequality of Askey and Gasper on Jacobi polynomials. The main reason lies in their wide applicability in areas such as Pade approximations, continued fractions, Tauberian theorems, numerical analysis, probability theory, mathematical statistics, scattering theory, nuclear physics, solid state physics, digital signal processing, electrical engineering, theoretical chemistry and so forth. This was emphasized and convincingly demonstrated during the presentations by both the principal speakers and the invited special lecturers. The main subjects of our Advanced Study Institute included complex orthogonal polynomials, signal processing, the recursion method, combinatorial interpretations of orthogonal polynomials, computational problems, potential theory, Pade approximations, Julia sets, special functions, quantum groups, weighted approximations, orthogonal polynomials associated with root systems, matrix orthogonal polynomials, operator theory and group representations.
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