Verfügbarkeit: Classical and quantum orthogonal polynomials in one variable

Classical and quantum orthogonal polynomials in one variable:

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Bibliographische Detailangaben
Beteilige Person:	Ismail, Mourad E. H. 1944- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2009
Ausgabe:	1. paperback. ed.
Schriftenreihe:	Encyclopedia of mathematics and its applications 98
Schlagwörter:	Orthogonale Polynome
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020497379&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020497379&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Includes bibliographical references (p. 663-698) and index
Umfang:	XVIII, 708 S.
ISBN:	9780521143479 9780521782012

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Datensatz im Suchindex

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adam_text	Contents Foreword page xi Preface xvi 1 Preliminaries 1 1.1 Hermitian Matrices and Quadratic Forms 1 1.2 Some Real and Complex Analysis 3 1.3 Some Special Functions 8 1.4 Summation Theorems and Transformations 12 2 Orthogonal Polynomials 16 2.1 Construction of Orthogonal Polynomials 16 2.2 Recurrence Relations 22 2.3 Numerator Polynomials 26 2.4 Quadrature Formulas 28 2.5 The Spectral Theorem 30 2.6 Continued Fractions 35 2.7 Modifications of Measures: Christoffel and Uvarov 37 2.8 Modifications of Measures: Toda 41 2.9 Modification by Adding Finite Discrete Parts 43 2.10 Modifications of Recursion Coefficients 45 2.11 DualSystems 47 3 Differential Equations, Discriminants and Electrostatics 52 3.1 Preliminaries 52 3.2 Differential Equations 53 3.3 Applications 63 3.4 Discriminants 67 3.5 An Electrostatic Equilibrium Problem 70 3.6 Functions of the Second Kind 73 3.7 Lie Algebras 76 4 Jacobi Polynomials 80 4.1 Orthogonality 80 4.2 Differential and Recursion Formulas 82 4.3 Generating Functions 88 4.4 Functions of the Second Kind 93 vi Contenti 4.5 Ultraspherical Polynomials 94 4.6 Laguerre and Hermite Polynomials 98 4.7 Multilinear Generating Functions 106 4.8 Asymptotics and Expansions 114 4.9 Relative Extrema of Classical Polynomials 120 4.10 The Bessel Polynomials 123 5 Some Inverse Problems 133 5.1 Ultraspherical Polynomials 1 3 З 5.2 Birth and Death Processes 136 5.3 The Hadamard Integral 141 5.4 Pollaczek Polynomials 147 5.5 A Generalization 151 5.6 Associated Laguerre and Hermite Polynomials 158 5.7 Associated Jacobi Polynomials 162 5.8 The J-Matrix Method 168 5.9 The Meixner-Pollaczek Polynomials 171 6 Discrete Orthogonal Polynomials 174 6.1 Meixner Polynomials 174 6.2 Hahn, Dual Hahn, and Krawtchouk Polynomials 177 6.3 Difference Equations 186 6.4 Discrete Discriminants 190 6.5 Lömmel Polynomials 194 6.6 An Inverse Operator 199 7 Zeros and Inequalities 203 7.1 A Theorem of Markov 203 7.2 Chain Sequences 205 7.3 The Hellmann-Feynman Theorem 211 7.4 Extreme Zeros of Orthogonal Polynomials 219 7.5 Concluding Remarks 221 8 Polynomials Orthogonal on the Unit Circle 222 8.1 Elementary Properties 222 8.2 Recurrence Relations 225 8.3 Differential Equations 231 8.4 Functional Equations and Zeros 240 8.5 Limit Theorems 245 8.6 Modifications of Measures 247 9 Linearization, Connections and Integral Representations 254 9.1 Connection Coefficients 256 9.2 The Ultraspherical Polynomials and Watson s Theorem 262 9.3 Linearization and Power Series Coefficients 264 9.4 Linearization of Products and Enumeration 269 9.5 Representations for Jacobi Polynomials 274 9.6 Addition and Product Formulas 277 9.7 The Askey-Gasper Inequality 281 Contents vii 10 The Sheffer Classification 283 10.1 Preliminaries 283 10.2 Delta Operators 286 10.3 Algebraic Theory 288 11 g-Series Preliminaries 294 11.1 Introduction 294 11.2 Orthogonal Polynomials 294 11.3 The Bootstrap Method 295 11.4 (/-Differences 297 12 (/-Summation Theorems 300 12.1 Basic Definitions 300 12.2 Expansion Theorems 303 12.3 Bilateral Series 308 12.4 Transformations 311 12.5 Additional Transformations 314 12.6 Theta Functions 316 13 Some g-Orthogonal Polynomials 319 13.1 ç-Hermite Polynomials 320 13.2 g-Ultraspherical Polynomials 327 13.3 Linearization and Connection Coefficients 331 13.4 Asymptotics 335 13.5 Application: The Rogers-Ramanujan Identities 336 13.6 Related Orthogonal Polynomials 341 13.7 Three Systems of g-Orthogonal Polynomials 345 14 Exponential and q-Bessel Functions 352 14.1 Definitions 352 14.2 Generating Functions 357 14.3 Addition Formulas 359 14.4 ρ -Analogues of Lömmel and Bessel Polynomials 360 14.5 A Class of Orthogonal Functions 364 14.6 An Operator Calculus 366 14.7 Polynomials of ç-Binomial Type 372 14.8 Another g-Umbral Calculus 376 15 The Askey-Wilson Polynomials 378 15.1 The Al-Salam-Chihara Polynomials 378 15.2 The Askey-Wilson Polynomials 382 15.3 Remarks 387 15.4 Asymptotics 389 15.5 Continuous q-Jacobi Polynomials and Discriminants 391 15.6 g-Racah Polynomials 396 15.7 (/-Integral Representations 400 15.8 Linear and Multilinear Generating Functions 405 15.9 Associated ç-Ultraspherical Polynomials 411 15.10 Two Systems of Orthogonal Polynomials 416 viii Contents 16 The Askey-Wilson Operators 426 16.1 Basic Results 426 16.2 A q-Sturm-Liouville Operator 433 16.3 The Askey-Wilson Polynomials 437 16.4 Connection Coefficients 443 16.5 Bethe Ansatz Equations of XXZ Model 446 17 g-Hermite Polynomials on the Unit Circle 455 17.1 The Rogers-Szegő Polynomials 455 17.2 Generalizations 460 17.3 q-Difference Equations 464 18 Discrete q-Orthogonal Polynomials 469 18.1 Discrete Sturm-Liouville Problems 469 18.2 The Al-Salam-Carlitz Polynomials 470 18.3 The Al-Salam-Carlitz Moment Problem 476 18.4 q-Jacobi Polynomials 477 18.5 q-Hahn Polynomials 484 18.6 (/-Differences and Quantized Discriminants 486 18.7 A Family of Biorthogonal Rational Functions 488 19 Fractional and q-Fractional Calculus 491 19.1 The Riemann-Liouville Operators 491 19.2 Bilinear Formulas 495 19.3 Examples 496 19.4 q-Fractional Calculus 501 19.5 Some Integral Operators 504 20 Polynomial Solutions to Functional Equations 509 20.1 Bochner s Theorem 509 20.2 Difference and g-Difference Equations 514 20.3 Equations in the Askey-Wilson Operators 516 20.4 Leonard Pairs and the q-Racah Polynomials 518 20.5 Characterization Theorems 525 21 Some Indeterminate Moment Problems 530 21.1 The Hamburger Moment Problem 530 21.2 A System of Orthogonal Polynomials 534 21.3 Generating Functions 537 21.4 The Nevanlinna Matrix 542 21.5 Some Orthogonality Measures 544 21.6 Ladder Operators 547 21.7 Zeros 550 21.8 The t/-Laguerre Moment Problem 553 21.9 Other Indeterminate Moment Problems 563 21.10 Some Biorthogonal Rational Functions 572 22 The Riemann-Hilbert Problem for Orthogonal Polynomials 578 22.1 The Cauchy Transform 578 Contents ix 22.2 The Fokas-Its-Kitaev Boundary Value Problem 581 22.2.1 The three-term recurrence relation 584 22.3 Hermite Polynomials 586 22.3.1 A Differential Equation 586 22.4 Laguerre Polynomials 589 22.4.1 Three-term recurrence relation 591 22.4.2 A differential equation 592 22.5 Jacobi Polynomials 596 22.5.1 Differential equation 597 22.6 Asymptotic Behavior 601 22.7 Discrete Orthogonal Polynomials 603 22.8 Exponential Weights 604 23 Multiple Orthogonal Polynomials 607 23.1 Type I and II Multiple Orthogonal Polynomials 608 23.1.1 Angelesco systems 610 23.1.2 AT systems 611 23.1.3 Biorthogonality 613 23.1.4 Recurrence relations 614 23.2 Hermite-Padé Approximation 621 23.3 Multiple Jacobi Polynomials 622 23.3.1 Jacobi-Angelesco polynomials 622 23.3.2 Jacobi-Piñeiro polynomials 626 23.4 Multiple Laguerre Polynomials 628 23.4.1 Multiple Laguerre polynomials of the first kind 628 23.4.2 Multiple Laguerre polynomials of the second kind 629 23.5 Multiple Hermite Polynomials 630 23.5.1 Random matrices with external source 631 23.6 Discrete Multiple Orthogonal Polynomials 632 23.6.1 Multiple Charlier polynomials 632 23.6.2 Multiple Meixner polynomials 632 23.6.3 Multiple Krawtchouk polynomials 634 23.6.4 Multiple Hahn polynomials 634 23.6.5 Multiple little g-Jacobi polynomials 635 23.7 Modified Bessel Function Weights 636 23.7.1 Modified Bessel functions 637 23.8 The Riemann-Hilbert Problem for Multiple Orthogonal Poly¬ nomials 639 23.8.1 Recurrence relation 644 23.8.2 Differential equation for multiple Hermite polynomials 645 24 Research Problems 648 24.1 Multiple Orthogonal Polynomials 648 24.2 A Class of Orthogonal Functions 649 24.3 Positivity 649 24.4 Asymptotics and Moment Problems 650 χ Contents 24.5 Functional Equations and Lie Algebras 652 24.6 Rogers-Ramanujan Identities 653 24.7 Characterization Theorems 654 24.8 Special Systems of Orthogonal Polynomials 658 24.9 Zeros of Orthogonal Polynomials 661 Bibliography 663 Index 699 Author index 705 Encyclopedia of Mathematics and its Applications Editorial Board P. Flajolet, M.E.H. Ismail, E. Lutwak Volume 98 Classical and Quantum Orthogonal Polynomials in One Variable This is first modern treatment of orthogonal polynomials from the viewpoint of special functions. The coverage is encyclopedic, including classical topics such as Jacobi, Hermite, Laguerre, Hahn, Charlier and Meixner polynomials as well as those, e.g. Askey-Wilson and Al-Salam-Chihara polynomial systems, discovered over the last 50 years: multiple orthogonal polynomials are dicussed for the first time in book form. Many modern applications of the subject are dealt with, including birth and death processes, integrable systems, combinatorics, and physical models. A chapter on open research problems and conjectures is designed to stimulate further research on the subject. Exercises of varying degrees of difficulty are included to help the graduate student and the newcomer. A comprehensive bibliography rounds off the work, which will be valued as an authoritative reference and for graduate teaching, in which role it has already been successfully class-tested.
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spellingShingle	Ismail, Mourad E. H. 1944- Classical and quantum orthogonal polynomials in one variable Encyclopedia of mathematics and its applications Orthogonale Polynome (DE-588)4172863-4 gnd
subject_GND	(DE-588)4172863-4
title	Classical and quantum orthogonal polynomials in one variable
title_auth	Classical and quantum orthogonal polynomials in one variable
title_exact_search	Classical and quantum orthogonal polynomials in one variable
title_full	Classical and quantum orthogonal polynomials in one variable Mourad E.H. Ismail ; with two chapters by Walter Van Assche
title_fullStr	Classical and quantum orthogonal polynomials in one variable Mourad E.H. Ismail ; with two chapters by Walter Van Assche
title_full_unstemmed	Classical and quantum orthogonal polynomials in one variable Mourad E.H. Ismail ; with two chapters by Walter Van Assche
title_short	Classical and quantum orthogonal polynomials in one variable
title_sort	classical and quantum orthogonal polynomials in one variable
topic	Orthogonale Polynome (DE-588)4172863-4 gnd
topic_facet	Orthogonale Polynome
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