Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 1 - Univariate Orthogonal Polynomials

401 Pages·13.859 MB·English

by  Mourad E. H. Ismail

ENCYCLOPEDIAOFSPECIALFUNCTIONS:
THEASKEY–BATEMANPROJECT
Volume1:UnivariateOrthogonalPolynomials
ThisisthefirstofthreevolumesthatformtheEncyclopediaofSpecialFunctions,anextensive
updateoftheBatemanManuscriptProject.
Volume 1 contains most of the material on orthogonal polynomials, from the classical
orthogonalpolynomialsofHermite,LaguerreandJacobitotheAskey–Wilsonpolynomials,
which are the most general basic hypergeometric orthogonal polynomials. Separate chap-
ters cover orthogonal polynomials on the unit circle, zeros of orthogonal polynomials and
matrixorthogonalpolynomials,withdetailedresultsaboutmatrix-valuedJacobipolynomials.
Achapteronmomentproblemsprovidesmanyexamplesofindeterminatemomentproblems.
Athoroughbibliographyroundsoffwhatwillbeanessentialreference.
mourade.h.ismail isResearchProfessorattheUniversityofCentralFlorida.

ENCYCLOPEDIA OF SPECIAL FUNCTIONS:
THE ASKEY–BATEMAN PROJECT
Volume 1: Univariate Orthogonal Polynomials
Editedby
MOURADE.H.ISMAIL
UniversityofCentralFlorida
withassistanceby
WALTERVANASSCHE
KULeuven,Belgium
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Informationonthistitle:www.cambridge.org/9780521197427
DOI:10.1017/9780511979156
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Title:Encyclopediaofspecialfunctions:theAskey-Batemanproject.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2020-|
Includesbibliographicalreferencesandindex.|Contents:VolumeI.
Univariateorthogonalpolynomials/editedbyMouradE.H.Mourad--
Identifiers:LCCN2020007276|ISBN9780521197427(hardback;v.1)
Subjects:LCSH:Functions,Special--Encyclopedias.
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ISBN–3VolumeSet978-1-108-88244-6Hardback
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Contents
Listofcontributors pageix
Preface xi
1 Preliminaries 1
1.1 AnalyticFacts 1
1.2 HypergeometricFunctions 2
1.3 SummationTheoremsandTransformations 7
1.4 q-Series 8
1.5 ThetaFunctions 15
1.6 Orthogonality 15
2 GeneralOrthogonalPolynomials 16
2.1 BasicFacts 16
2.2 NumeratorsandQuadratures 22
2.3 TheSpectralTheorem 24
2.4 ContinuedFractions 27
2.5 ModificationsofMeasuresandRecursions 30
2.6 LinearizationandConnectionRelations 34
2.7 AdditionTheorems 37
2.8 DifferentialEquations 39
2.9 DiscriminantsandElectrostatics 44
2.10 FunctionsoftheSecondKind 47
2.11 DualSystems 48
2.12 MomentRepresentationsandDeterminants 50
3 JacobiandRelatedPolynomials 51
3.1 RecursionsandRepresentations 51
3.2 GeneratingFunctions 56
3.3 JacobiFunctionsoftheSecondKind 59
3.4 Routh–JacobiPolynomials 63
3.5 Ultraspherical(Gegenbauer)Polynomials 63
3.6 ChebyshevPolynomials 67
vi Contents
3.7 LegendrePolynomials 71
3.8 LaguerreandHermitePolynomials 74
3.9 TheComplexHermitePolynomials 82
3.10 HermiteFunctions 84
3.11 MultilinearGeneratingFunctions 85
3.12 IntegralRepresentations 88
3.13 Asymptotics 90
3.14 RelativeExtremaofClassicalPolynomials 95
3.15 TheBesselPolynomials 96
4 RecursivelyDefinedPolynomials 100
4.1 BirthandDeathProcessPolynomials 100
4.2 PolynomialsofPollaczekType 102
4.3 AssociatedLaguerreandHermitePolynomials 108
4.4 AssociatedJacobiPolynomials 109
4.5 SievedPolynomials 114
5 WilsonandRelatedPolynomials 119
5.1 TheMeixner–PollaczekPolynomials 119
5.2 WilsonPolynomials 121
5.3 ContinuousDualHahnPolynomials 124
5.4 ContinuousHahnPolynomials 126
6 DiscreteOrthogonalPolynomials 129
6.1 MeixnerandCharlierPolynomials 129
6.2 Hahn,DualHahn,andKrawtchoukPolynomials 131
6.3 DifferenceEquations 138
6.4 LommelPolynomialsandRelatedPolynomials 141
6.5 AnInverseOperator 145
6.6 q-Sturm–LiouvilleProblems 146
6.7 TheAl-Salam–CarlitzPolynomials 148
6.8 q-JacobiPolynomials 151
6.9 q-HahnPolynomials 154
6.10 AFamilyofBiorthogonalRationalFunctions 156
7 Someq-OrthogonalPolynomials 157
7.1 q-HermitePolynomials 157
7.2 q-UltrasphericalPolynomials 161
7.3 Asymptotics 168
7.4 IntegralsandtheRogers–RamanujanIdentities 168
7.5 AGeneralizationoftheSchurPolynomials 170
7.6 Associatedq-UltrasphericalPolynomials 172
7.7 TwoSystemsofq-OrthogonalPolynomials 175
Contents vii
8 TheAskey–WilsonFamilyofPolynomials 178
8.1 Al-Salam–ChiharaPolynomials 178
8.2 TheAskey–WilsonPolynomials 181
8.3 TheAskey–WilsonEquation 185
8.4 Continuousq-JacobiPolynomialsandDiscriminants 187
8.5 q-RacahPolynomials 190
8.6 LinearandMultilinearGeneratingFunctions 192
8.7 AssociatedAskey–WilsonPolynomials 195
9 OrthogonalPolynomialsontheUnitCircle 199
L.Golinskii
9.1 DefinitionsandBasicProperties 199
9.2 Szego˝ RecurrenceRelationsandVerblunskyCoefficients 202
9.3 Szego˝’sTheoryandItsExtensions 207
9.4 ZerosofOPUC 217
9.5 CMVMatrices–UnitaryAnaloguesofJacobiMatrices 221
9.6 DifferentialEquations 224
9.7 ExamplesofOPUC 226
9.8 ModificationofMeasures 231
10 ZerosofOrthogonalPolynomials 242
A.Laforgia&M.Muldoon
10.1 Introduction 242
10.2 GeneralResultsonZeros 242
10.3 JacobiPolynomials 249
10.4 UltrasphericalPolynomials 255
10.5 LegendrePolynomials 259
10.6 LaguerrePolynomials 260
10.7 HermitePolynomialsandFunctions 264
10.8 OtherOrthogonalPolynomials 267
11 TheMomentProblem 269
C.Berg&J.S.Christiansen
11.1 HamburgerMomentProblems 269
11.2 StieltjesMomentProblems 280
11.3 ExamplesofIndeterminateMomentProblems 285
12 Matrix-ValuedOrthogonalPolynomialsandDifferentialEquations 307
A.Dura´n&F.A.Gru¨nbaum
12.1 MatrixPolynomialsandMatrixOrthogonality 307
12.2 Matrix-Valued Orthogonal Polynomials Satisfying Second-Order
DifferentialEquations 315
viii Contents
13 SomeFamiliesofMatrix-ValuedJacobiOrthogonalPolynomials 334
F.A.Gru¨nbaum,I.Pacharoni,&J.A.Tirao
13.1 Introduction 334
13.2 SphericalFunctions 334
13.3 Matrix-ValuedSphericalFunctionsAssociatedtoP (C) 337
2
13.4 TheSphericalFunctionsasMatrix-ValuedHypergeometricFunctions 340
13.5 MatrixOrthogonalPolynomialsArisingfromSphericalFunctions 344
13.6 TheMatrixJacobiPolynomialsArisingfromP (C) 348
d
13.7 Miscellanea 354
References 357
Index 385
Contributors
LeonidGolinskii InstituteforLowTemperaturePhysicsandEngineering,UkrainianAcad-
emyofSciences,47LeninAvenue,Kharkov61103,Ukraine
AndreaLaforgia DepartmentofMathematics,Universita` degliStudiRomaTre,LargoSan
LeonardoMurialdo1,IT–00146Rome,Italy.
Martin Muldoon Department of Mathematics & Statistics, York University, Toronto, On-
tarioM3J1P3,Canada.
Christian Berg Institute for Mathematical Sciences, University of Copenhagen, DK–2100
Copenhagen,Denmark.
Jacob S. Christiansen Lund University, Centre for Mathematical Sciences, Box 118, SE–
22100,Lund,Sweden.
Antonio J. Dura´n Departamento de Ana´lisis Matema´tico, Universidad de Sevilla, Apdo
(P.O.Box)1160,ES–41080Sevilla,Spain.
F.AlbertoGru¨nbaum DepartmentofMathematics,UniversityofCalifornia,Berkeley,CA
94720,USA.
InesPacharoni CIEM-FaMAF,UniversidadNacionaldeCo´rdoba,5000Co´rdoba,Argentina
JuanA.Tirao CIEM-FaMAF,UniversidadNacionaldeCo´rdoba,5000Co´rdoba,Argentina