Askey Theory and Application of Special Functions

Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin-Madison, March 31-April 2, 1975
1. Auflage 2014
ISBN: 978-1-4832-1616-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 572 Seiten, Web PDF

Theory and Application of Special Functions contains the proceedings of the Advanced Seminar on Special Functions sponsored by the Mathematics Research Center of the University of Wisconsin-Madison and held from March 31 to April 2, 1975. The seminar tackled the theory and application of special functions and covered topics ranging from the asymptotic estimation of special functions to association schemes and coding theory. Some interesting results, conjectures, and problems are given. Comprised of 13 chapters, this book begins with a survey of computational methods in special functions, followed by a discussion on unsolved problems in the asymptotic estimation of special functions. The reader is then introduced to periodic Bernoulli numbers, summation formulas, and applications; problems and prospects for basic hypergeometric functions; and linear growth models with many types and multidimensional Hahn polynomials. Subsequent chapters explore two-variable analogues of the classical orthogonal polynomials; special functions of matrix and single argument in statistics; and some properties of the determinants of orthogonal polynomials. This monograph is intended primarily for students and practitioners of mathematics.

Autoren/Hrsg.

Askey, Richard

Inhaltsverzeichnis

1;Front Cover;1
2;Theory and Application of Special Functions;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;8
6;Preface;10
7;Chapter 1. Computational Methods in Special Functions-A Survey;14
7.1;Introduction;16
7.2;1. Methods based on preliminary approximation;17
7.3;2. Methods based on linear recurrence relations;53
7.4;3. Nonlinear recurrence algorithms for elliptic integrals and elliptic functions;64
7.5;4. Computer software for special functions;80
7.6;REFERENCES;83
8;Chapter 2. Unsolved Problems in the Asymptotic Estimation of Special Functions;112
8.1;Abstract;112
8.2;1. INTRODUCTION;113
8.3;PART I. THE TOOLS OF ASYMPTOTIC ANALYSIS;117
8.3.1;2. INTEGRALS;117
8.3.2;3. SUMS AND SEQUENCES;123
8.3.3;4. LINEAR ORDINARY DIFFERENTIAL EQUATIONS;126
8.4;PART II. ASYMPTOTIC ESTIMATES OF THE SPECIAL FUNCTIONS;132
8.4.1;5. FUNCTIONS OF ONE OR TWO VARIABLES;132
8.4.2;6. FUNCTIONS OF THREE VARIABLES;135
8.4.3;7. FUNCTIONS OF FOUR OR MORE VARIABLES;146
8.4.4;ACKNOWLEDGMENTS;149
8.4.5;REFERENCES;149
9;Chapter 3. Periodic Bernoulli Numbers, Summation Formulas and Applications;156
9.1;1. Introduction.;156
9.2;2. Periodic Bernoulli numbers and polynomials;156
9.3;3. The periodic Poisson and periodic Euler-Maclaurin summation;161
9.4;4. The distribution of quadratic residues;163
9.5;5. Power sums and cotangent sums;167
9.6;6. Gauss sums;169
9.7;7. Functional equations;170
9.8;8. A trigonometric series of Hardy and Littlewood;171
9.9;9. Infinite series of ordinary Bessel functions;176
9.10;10. Infinite series of modified Bessel functions;184
9.11;11. Entries from Ramanujan's Notebooks and kindred formulae;186
9.12;REFERENCES;196
10;Chapter 4. Problems and Prospects for Basic Hypergeometric Functions;204
10.1;1. Introduction;204
10.2;2. Partitions identities;205
10.3;3. Identities for Multiple Hypergeometric Series;211
10.4;4. Basic Appell and Lauricella Series;219
10.5;5. MacMahon's Master Theorem and the Dyson Conjecture;226
10.6;6. Saalschützian Series and Inversion Theorems;231
10.7;7. Conclusion.;233
10.8;REFERENCES;234
11;Chapter 5. An Introduction to Association Schemes and Coding Theory;238
11.1;ABSTRACT;238
11.2;1 INTRODUCTION;238
11.3;2 Error-Correcting Codes;240
11.4;3 Association Schemes;244
11.5;4 The Hamming Association Scheme;251
11.6;5 The Johnson Association Scheme;256
11.7;6 Association Schemes Obtained from Graphs and Other Sources;257
11.8;7 The Linear Programming Bound;260
11.9;8 Properties of Perfect Codes;267
11.10;REFERENCES;268
12;Chapter 6. Linear Growth Models with Many Types and Multidimensional Hahn Polynomials;274
12.1;1. Multi-allele Moran mutation models;274
12.2;2. Representation of P(t).;278
12.3;3. Relation with multi-dimensional linear growth;280
12.4;4. The case r = 2 and the Hahn polynomials;285
12.5;5. Moran model with r types.;289
12.6;6. Linear growth model with r types;297
12.7;7. The eigenfunctions when;299
12.8;REFERENCES;301
13;Chapter 7. Orthogonal Polynomials Revisited;302
13.1;I. Introduction;302
13.2;II. Polynomials on the Real Axis;303
13.3;III. Applications;309
13.4;IV. Polynomials on the Unit Circle;312
13.5;V. Conclusion;317
13.6;FOOTNOTES;317
14;Chapter 8. Symmetry, Separation of Variables, and Special Functions;318
14.1;REFERENCES;360
15;Chapter 9. Nicholson-Type Integrals for Products of Gegenbauer Functions and Related Topics;366
15.1;ABSTRACT;366
15.2;1. INTRODUCTION;367
15.3;2. DERIVATION OF A NICHOLSON-TYPE FORMULA FOR GEGENBAUER FUNCTIONS;368
15.4;3. SOME APPLICATIONS FOR GEGENBAUER FUNCTIONS;374
15.5;4. DEDUCTIONS FOR OTHER FUNCTIONS;381
15.6;5. CONCLUSIONS;385
15.7;FOOTNOTES;386
15.8;REFERENCES;386
16;Chapter 10. Positivity and Special Functions;388
16.1;1. Introduction;388
16.2;2. Background material.;390
16.3;3. Computation.;393
16.4;4. Recurrence relations.;410
16.5;5. Maximum principles.;415
16.6;6. Estimation.;417
16.7;7. Absolutely monotonie and completely monotonie functions;420
16.8;8. Sums of squares.;423
16.9;9. Additional comments.;432
16.10;REFERENCES;437
17;Chapter 11. Two-Variable Analogues of the Classical Orthogonal Polynomials;448
17.1;1. Introduction;449
17.2;2. Jacobi polynomials;450
17.3;3. Methods of constructing two-variable analogues of the Jacobipolynomials;457
17.4;4. Differential recurrence relations and other analytic properties;483
17.5;5. Orthogonal polynomials in two variables as spherical functions;497
17.6;REFERENCES;502
18;Chapter 12. Special Functions of Matrix and Single Argument in Statistics;510
18.1;FOREWORD;510
18.2;INTRODUCTION;511
18.3;I. Zonal Polynomials of the Real Positive Definite Symmetric Matrices;511
18.4;II. Orthogonal Polynomials of Matrix Argument;518
18.5;III. Differential Equations for Hypergeometric Functions of Matrix Argument;523
18.6;IV. Probability Functions of Ordered Roots;525
18.7;REFERENCES;531
19;Chapter 13. Some Properties of Determinants of Orthogonal Polynomials;534
19.1;Introduction;534
19.2;1. Recurrence formulas;535
19.3;2. CHRISTQFFEL-DAJRBOUX FORMULA AND WRONSKIAN IDENTITIES.;543
19.4;3. Further Properties;559
19.5;REFERENCES;562
20;Index;564