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Lectures on Orthogonal Polynomials and Special Functions [Minkštas viršelis]

Edited by , Edited by (University of Central Florida)
  • Formatas: Paperback / softback, 350 pages, aukštis x plotis x storis: 228x151x20 mm, weight: 520 g, Worked examples or Exercises
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 15-Oct-2020
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108821596
  • ISBN-13: 9781108821599
Kitos knygos pagal šią temą:
Lectures on Orthogonal Polynomials and Special FunctionsDidesnis paveikslėlis
  • Formatas: Paperback / softback, 350 pages, aukštis x plotis x storis: 228x151x20 mm, weight: 520 g, Worked examples or Exercises
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 15-Oct-2020
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108821596
  • ISBN-13: 9781108821599
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781108821599.html
Raktažodžiai:
Written by experts in their respective fields, this collection of pedagogic surveys provides detailed insight and background into five separate areas at the forefront of modern research in orthogonal polynomials and special functions at a level suited to graduate students. A broad range of topics are introduced including exceptional orthogonal polynomials, q-series, applications of spectral theory to special functions, elliptic hypergeometric functions, and combinatorics of orthogonal polynomials. Exercises, examples and some open problems are provided. The volume is derived from lectures presented at the OPSF-S6 Summer School at the University of Maryland, and has been carefully edited to provide a coherent and consistent entry point for graduate students and newcomers.

Daugiau informacijos

Contains graduate-level introductions by international experts to five areas of research in orthogonal polynomials and special functions.
Contributors x
Preface xi
1 Exceptional Orthogonal Polynomials via Krai) Discrete Polynomials Antonio J. Duran
1(75)
1.1 Background on classical and classical discrete polynomials
4(14)
1.1.1 Weights on the real line
4(1)
1.1.2 The three-term recurrence relation
5(1)
1.1.3 The classical orthogonal polynomial families
6(4)
1.1.4 Second-order differential operator
10(2)
1.1.5 Characterizations of the classical families of orthogonal polynomials
12(1)
1.1.6 The classical families and the basic quantum models
13(2)
1.1.7 The classical discrete families
15(3)
1.2 The Askey tableau. Krall and exceptional polynomials. Darboux Transforms
18(12)
1.2.1 The Askey tableau
18(3)
1.2.2 Krall and exceptional polynomials
21(2)
1.2.3 Krall polynomials
23(2)
1.2.4 Darboux transforms
25(5)
1.3 D-operators
30(8)
1.3.1 D-operators
30(2)
1.3.2 D-operators on the stage
32(5)
1.3.3 D-operators of type 2
37(1)
1.4 Constructing Krall polynomials by using D-operators
38(10)
1.4.1 Back to the orthogonality
39(1)
1.4.2 Krall-Laguerre polynomials
40(2)
1.4.3 Krall discrete polynomials
42(6)
1.5 First expansion of the Askey tableau. Exceptional polynomials: discrete case
48(12)
1.5.1 Comparing the Krall continuous and discrete cases (roughly speaking): Darboux transform
48(2)
1.5.2 First expansion of the Askey tableau
50(3)
1.5.3 Exceptional polynomials
53(3)
1.5.4 Constructing exceptional discrete polynomials by using duality
56(4)
1.6 Exceptional polynomials: continuous case. Second expansion of the Askey tableau
60(8)
1.6.1 Exceptional Charlier polynomials: admissibility
60(2)
1.6.2 Exceptional Hermite polynomials by passing to the limit
62(2)
1.6.3 Exceptional Meixner and Laguerre polynomials
64(4)
1.6.4 Second expansion of the Askey tableau
68(1)
1.7 Appendix: Symmetries for Wronskian type determinants whose entries are classical and classical discrete orthogonal polynomials
68(8)
References
70(6)
2 A Brief Review of q-Series Mourad E. M. Ismail
76(55)
2.1 Introduction
76(1)
2.2 Notation and q-operators
77(4)
2.3 q-Taylor series
81(4)
2.4 Summation theorems
85(5)
2.5 Transformations
90(4)
2.6 q-Hermite polynomials
94(7)
2.7 The Askey-Wilson polynomials
101(5)
2.8 Ladder operators and Rodrigues formulas
106(7)
2.9 Identities and summation theorems
113(2)
2.10 Expansions
115(4)
2.11 Askey-Wilson expansions
119(6)
2.12 A q-exponential function
125(6)
References
128(3)
3 Applications of Spectral Theory to Special Functions Erik Koelink
131(82)
3.1 Introduction
133(6)
3.2 Three-term recurrences in 12(z)
139(10)
3.2.1 Exercises
147(2)
3.3 Three-term recurrence relations and orthogonal polynomials
149(6)
3.3.1 Orthogonal polynomials
149(3)
3.3.2 Jacobi operators
152(2)
3.3.3 Moment problems
154(1)
3.3.4 Exercises
154(1)
3.4 Matrix-valued orthogonal polynomials
155(19)
3.4.1 Matrix-valued measures and related polynomials
156(7)
3.4.2 The corresponding Jacobi operator
163(4)
3.4.3 The resolvent operator
167(4)
3.4.4 The spectral measure
171(2)
3.4.5 Exercises
173(1)
3.5 More on matrix weights, matrix-valued orthogonal polynomials and Jacobi operators
174(6)
3.5.1 Matrix weights
174(2)
3.5.2 Matrix-valued orthogonal polynomials
176(2)
3.5.3 Link to case of l2 (Z)
178(1)
3.5.4 Reducibility
179(1)
3.5.5 Exercises
180(1)
3.6 The/-matrix method
180(19)
3.6.1 Schrodinger equation with the Morse potential
182(4)
3.6.2 A tridiagonal differential operator
186(4)
3.6.3 J-matrix method with matrix-valued orthogonal polynomials
190(8)
3.6.4 Exercises
198(1)
3.7 Appendix: The spectral theorem
199(7)
3.7.1 Hilbert spaces and operators
199(2)
3.7.2 Hilbert C*-modules
201(1)
3.7.3 Unbounded operators
202(1)
3.7.4 The spectral theorem for bounded self-adjoint operators
202(2)
3.7.5 Unbounded self-adjoint operators
204(1)
3.7.6 The spectral theorem for unbounded self-adjoint operators
205(1)
3.8 Hints and answers for selected exercises
206(7)
References
207(6)
4 Elliptic Hypergeometric Functions Hjalmar Rosengren
213(67)
4.1 Elliptic functions
216(19)
4.1.1 Definitions
216(1)
4.1.2 Theta functions
217(3)
4.1.3 Factorization of elliptic functions
220(2)
4.1.4 The three-term identity
222(1)
4.1.5 Even elliptic functions
223(3)
4.1.6 Interpolation and partial fractions
226(3)
4.1.7 Modularity and elliptic curves
229(4)
4.1.8 Comparison with classical notation
233(2)
4.2 Elliptic hypergeometric functions
235(27)
4.2.1 Three levels of hypergeometry
235(2)
4.2.2 Elliptic hypergeometric sums
237(2)
4.2.3 The Frenkel-Turaev sum
239(4)
4.2.4 Well-poised and very well-poised sums
243(2)
4.2.5 Thesum 12V11
245(3)
4.2.6 Biorthogonal rational functions
248(2)
4.2.7 A quadratic summation
250(3)
4.2.8 An elliptic Minton summation
253(2)
4.2.9 The elliptic gamma function
255(1)
4.2.10 Elliptic hypergeometric integrals
256(2)
4.2.11 Spiridonov's elliptic beta integral
258(4)
4.3 Solvable lattice models
262(18)
4.3.1 Solid-on-solid models
262(3)
4.3.2 The Yang-Baxter equation
265(1)
4.3.3 The R-operator
266(2)
4.3.4 The elliptic SOS model
268(2)
4.3.5 Fusion and elliptic hypergeometry
270(6)
References
276(4)
5 Combinatorics of Orthogonal Polynomials and their Moments Jiang Zeng
280(51)
5.1 Introduction
280(3)
5.2 General and combinatorial theories of formal OPS
283(11)
5.2.1 Formal theory of orthogonal polynomials
283(8)
5.2.2 The Flajolet-Viennot combinatorial approach
291(3)
5.3 Combinatorics of generating functions
294(10)
5.3.1 Exponential formula and Foata's approach
294(3)
5.3.2 Models of orthogonal Sheffer polynomials
297(2)
5.3.3 MacMahon's Master Theorem and a Mehler-type formula
299(5)
5.4 Moments of orthogonal Sheffer polynomials
304(10)
5.4.1 Combinatorics of the moments
304(5)
5.4.2 Linearization coefficients of Sheffer polynomials
309(5)
5.5 Combinatorics of some q-polynomials
314(16)
5.5.1 Al-Salam-Chihara polynomials
314(1)
5.5.2 Moments of continuous q-Hermite, q-Charlier and q-Laguerre polynomials
315(3)
5.5.3 Linearization coefficients of continuous q-Hermite, q-Charlier and q-Laguerre polynomials
318(5)
5.5.4 A curious q-analogue of Hermite polynomials
323(5)
5.5.5 Combinatorics of continued fractions and y-positivity
328(2)
5.6 Some open problems
330(1)
References 331
Howard S. Cohl, mathematician in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology, is Technical Editor for the NIST Digital Library of Mathematical Functions. He has PhD's in mathematics and physics and has published over 60 papers on special functions, orthogonal polynomials, fundamental solutions for linear PDEs on Riemannian manifolds, and mathematical knowledge management. Mourad E. H. Ismail, Research Professor of Mathematics at the University of Central Florida, is an expert in special functions and approximation theory. He has made substantial contributions to the theory of orthogonal polynomials and q-series. Ismail is an editor of several research journals and author and editor of seven books.