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E-raamat: Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

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Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special FunctionsSuurem pilt
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The Encyclopedia of Special Functions provides an extensive update of the Bateman Manuscript Project. The three volumes will be indispensable for all scientists who use special functions in their research. Volume 2 provides detailed and up-to-date information on multivariable special functions.

This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

Arvustused

'Overall the volume is a very useful addition to any research mathematician's library who works on these topics.' Manjil Pratim Saikia, zbMATH

Muu info

The second volume of an extensive update of the Bateman Manuscript Project discusses multivariable special functions.
List of Contributors
ix
Preface xi
1 General Overview of Multivariable Special Functions
1(18)
T. H. Koornwinder
J. V. Stokman
1.1 Introduction
1(2)
1.2 Multivariable Classical, Basic and Elliptic Hypergeometric Series
3(7)
1.3 Multivariable (Bi)Orthogonal Polynomials and Functions
10(1)
1.4 Multivariable (Bi)Orthogonal Polynomials and Functions, Some Details
11(3)
References
14(5)
2 Orthogonal Polynomials of Several Variables
19(60)
Yuan Xu
2.1 Introduction
19(1)
2.2 General Properties of Orthogonal Polynomials of Several Variables
20(9)
2.3 Orthogonal Polynomials of Two Variables
29(13)
2.4 Spherical Harmonics
42(5)
2.5 Classical Orthogonal Polynomials of Several Variables
47(9)
2.6 Relation Between Orthogonal Polynomials on Classical Domains
56(3)
2.7 Orthogonal Expansions and Summability
59(2)
2.8 Discrete Orthogonal Polynomials of Several Variables
61(6)
2.9 Other Orthogonal Polynomials of Several Variables
67(5)
References
72(7)
3 Appell and Lauricella Hypergeometric Functions
79(22)
K. Matsumoto
3.1 Introduction
79(1)
3.2 Appell's Hypergeometric Series
80(1)
3.3 Lauricella's Hypergeometric Series
80(1)
3.4 Integral Representations
81(3)
3.5 Systems of Hypergeometric Differential Equations
84(5)
3.6 Local Solution Spaces
89(1)
3.7 Transformation Formulas
90(1)
3.8 Contiguity Relations
91(1)
3.9 Monodromy Representations
92(3)
3.10 Twisted Period Relations
95(1)
3.11 The Schwarz Map for Lauricella's Fq
96(1)
3.12 Reduction Formulas
97(1)
References
97(4)
4 A -Hypergeometric Functions
101(21)
N. Takayama
4.1 Introduction
101(1)
4.2 A-Hypergeometric Equations
101(5)
4.3 Combinatorics, Polytopes and Grobner Basis
106(1)
4.4 A -Hypergeometric Series
107(6)
4.5 Hypergeometric Function of Type E(k, n)
113(1)
4.6 Contiguity Relations
114(1)
4.7 Properties of A-Hypergeometric Equations
115(3)
4.8 A-Hypergeometric Polynomials and Statistics
118(1)
References
119(3)
5 Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems
122(37)
M. J. Schlosser
5.1 Introduction
122(3)
5.2 Some Identities for (Basic) Hypergeometric Series Associated with Root Systems
125(18)
5.3 Hypergeometric and Basic Hypergeometric Integrals Associated with Root Systems
143(3)
5.4 Basic Hypergeometric Series with Macdonald Polynomial Argument
146(5)
5.5 Remarks on Applications
151(1)
References
152(7)
6 Elliptic Hypergeometric Functions Associated with Root Systems
159(28)
H. Rosengren
S. O. Warnaar
6.1 Introduction
159(5)
6.2 Integrals
164(5)
6.3 Series
169(5)
6.4 Elliptic Macdonald-Koornwinder Theory
174(9)
References
183(4)
7 Dunkl Operators and Related Special Functions
187(30)
C. F. Dunkl
7.1 Introduction
187(1)
7.2 Root Systems
188(4)
7.3 Invariant Polynomials
192(1)
7.4 Dunkl Operators
193(7)
7.5 Harmonic Polynomials
200(3)
7.6 The Intertwining Operator and the Dunkl Kernel
203(5)
7.7 The Dunkl Transform
208(1)
7.8 The Poisson Kernel
209(1)
7.9 Harmonic Polynomials for R2
210(2)
7.10 Nonsymmetric Jack Polynomials
212(3)
References
215(2)
8 Jacobi Polynomials and Hypergeometric Functions Associated with Root Systems
217(41)
G. J. Heck man
E. M. Opdam
8.1 The Gauss Hypergeometric Function
217(2)
8.2 Root Systems
219(1)
8.3 The Hypergeometric System
220(4)
8.4 Jacobi Polynomials
224(8)
8.5 The Calogero-Moser System
232(4)
8.6 The Hypergeometric Function
236(12)
8.7 Special Cases
248(6)
References
254(4)
9 Macdonald-Koornwinder Polynomials
258(56)
J. V. Stokman
9.1 Introduction
258(7)
9.2 The Basic Representation of the Extended Affine Hecke Algebra
265(9)
9.3 Monic Macdonald-Koornwinder Polynomials
274(15)
9.4 Double Affine Hecke Algebras and Normalized Macdonald-Koornwinder Polynomials
289(11)
9.5 Explicit Evaluation and Norm Formulas
300(1)
A Appendix
301(8)
References
309(5)
10 Combinatorial Aspects of Macdonald and Related Polynomials
314(54)
J. Haglund
10.1 Introduction
314(1)
10.2 Basic Theory of Symmetric Functions
315(8)
10.3 Analytic and Algebraic Properties of Macdonald Polynomials
323(9)
10.4 The Combinatorics of the Space of Diagonal Harmonics
332(9)
10.5 The Expansion of the Macdonald Polynomial into Monomials
341(4)
10.6 Consequences of Theorem 10.5.3
345(5)
10.7 Nonsymmetric Macdonald Polynomials
350(5)
10.8 The Genesis of the q, /-Catalan Statistics
355(3)
10.9 Other Directions
358(1)
10.10 Recent Developments
359(2)
References
361(7)
11 Knizhnik-Zamolodchikov-Type Equations, Selberg Integrals and Related Special Functions
368(34)
V. Tarasov
A. Varchenko
11.1 Introduction
368(1)
11.2 Representation Theory
369(1)
11.3 Rational KZ Equation and Gaudin Model
370(3)
11.4 Hypergeometric Solutions of the Rational KZ and Dynamical Equations, and Bethe Ansatz
373(4)
11.5 Trigonometric KZ Equation
377(2)
11.6 Hypergeometric Solutions of the Trigonometric KZ Equation and Bethe Ansatz
379(2)
11.7 Knizhnik-Zamolodchikov-Bernard Equation and Elliptic Hypergeometric Functions
381(4)
11.8 qKZ Equation
385(3)
11.9 Hypergeometric Solutions of the qKZ Equations and Bethe Ansatz
388(4)
11.10 One-integration Examples
392(2)
11.11 Selberg-Type Integrals
394(2)
11.12 Further Development
396(1)
References
397(5)
12 9j-Coefficients and Higher
402(18)
J. Van der Jeugt
12.1 Introduction
402(1)
12.2 Representations of the Lie Algebra su(2)
403(1)
12.3 Clebsch-Gordan Coefficients and 3j-Coefficients
404(4)
12.4 Racah Coefficients and 6y-Coefficients
408(4)
12.5 The9 j-Coefficient
412(4)
12.6 Beyond 9j: Graphical Methods
416(1)
References
417(3)
Index 420
Tom H. Koornwinder is Professor Emeritus at the University of Amsterdam. He is an expert in special functions, orthogonal polynomials, and Lie theory. He introduced the five-parameter extension of the BC-type Macdonald polynomials, which are nowadays called Koornwinder polynomials. He was co-author of the chapter on orthogonal polynomials in the Digital Library of Mathematical Functions, and is involved in its revision. Jasper V. Stokman is Professor of Mathematics at the University of Amsterdam. He is an expert in special functions, Lie theory, and integrable systems. He introduced BC-type extensions of several families of classical orthogonal polynomials, and nonpolynomial generalizations of Koornwinder polynomials. He linked multivariable special functions to harmonic analysis on quantum groups and Hecke algebras, and to statistical mechanics and analytic number theory.
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