Classical and Quantum Orthogonal Polynomials in One Variable

726 Pages·2005·4.272 MB·English

by  Mourad E. H. Ismail

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
EditorialBoard
P.Flagolet,M.E.H.Ismail,E.Lutwak
Volume98
ClassicalandQuantumOrthogonalPolynomialsinOneVariable
Thisisfirstmoderntreatmentoforthogonalpolynomialsfromtheviewpoint
ofspecialfunctions. Thecoverageisencyclopedic,includingclassicaltopics
suchasJacobi,Hermite,Laguerre,Hahn,CharlierandMeixnerpolynomials
aswellasthose,e.g. Askey–WilsonandAl-Salam–Chiharapolynomial
systems,discoveredoverthelast50years: multipleorthogonalpolynomials
aredicussedforthefirsttimeinbookform. Manymodernapplicationsof
thesubjectaredealtwith,includingbirthanddeathprocesses,integrable
systems,combinatorics,andphysicalmodels. Achapteronopenresearch
problemsandconjecturesisdesignedtostimulatefurtherresearchonthe
subject.
Exercisesofvaryingdegreesofdifficultyareincludedtohelpthe
graduatestudentandthenewcomer. Acomprehensivebibliographyrounds
offthework,whichwillbevaluedasanauthoritativereferenceandfor
graduateteaching,inwhichroleithasalreadybeensuccessfullyclass-tested.
ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
http://publishing.cambridge.org/stm/mathematics/eom/
88. TeoMoraSolvingPolynomialEquationSystemsI
89. KlausBichtelerStochasticIntegrationwithJumps
90. M.LothaireAlgebraicCombinatoricsonWords
91. A.A.Ivanov&S.V.ShpectorovGeometryofSporadicGroups,2
92. PeterMcMullen&EgonSchulteAbstractRegularPolytopes
93. G.Gierzetal.ContinuousLatticesandDomains
94. StevenR.FinchMathematicalConstants
95. YoussefJabriTheMountainPassTheorem
96. GeorgeGasper&MizanRahmanBasicHypergeometricSeries2nded.
97. MariaCristinaPedicchio&WalterTholenCategoricalFoundations
98. MouradIsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99. TeoMoraSolvingPolynomialEquationSystemsII
100. EnzoOlivieri&MariaEulaliaVaresLargeDeviationsandMetastability
101. A.Kushner,V.Lychagin&V.RoubtsovContactGeometryandNonlinearDifferential
Equations
102. R.J.Wilson&L.BeinekeTopicsinAlgebraicGraphTheory
Classical and Quantum Orthogonal
Polynomials in One Variable
MouradE.H.Ismail
UniversityofCentralFlorida
Withtwochaptersby
WalterVanAssche
CatholicUniversityofLeuven
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo
CambridgeUniversityPress,
ThePittBuilding,TrumpingtonStreet,Cambridge,UnitedKingdom
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521782012
(cid:1)c CambridgeUniversityPress2005
Thisbookisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2005
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
TypefaceTimes10/13pt SystemLATEX2ε [AUTHOR]
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN-13 978–0–521–78201–2hardback
ISBN-10 0–521–78201–5hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLSforexternalor
third-partyinternetwebsitesreferredtointhisbook,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Foreword pagexi
Preface xvi
1 Preliminaries 1
1.1 HermitianMatricesandQuadraticForms 1
1.2 SomeRealandComplexAnalysis 3
1.3 SomeSpecialFunctions 8
1.4 SummationTheoremsandTransformations 12
2 OrthogonalPolynomials 16
2.1 ConstructionofOrthogonalPolynomials 16
2.2 RecurrenceRelations 22
2.3 NumeratorPolynomials 26
2.4 QuadratureFormulas 28
2.5 TheSpectralTheorem 30
2.6 ContinuedFractions 35
2.7 ModificationsofMeasures: ChristoffelandUvarov 37
2.8 ModificationsofMeasures: Toda 41
2.9 ModificationbyAddingFiniteDiscreteParts 43
2.10 ModificationsofRecursionCoefficients 45
2.11 DualSystems 47
3 DifferentialEquations 52
3.1 Preliminaries 52
3.2 DifferentialEquations 53
3.3 Applications 63
3.4 Discriminants 67
3.5 AnElectrostaticEquilibriumProblem 70
3.6 FunctionsoftheSecondKind 73
3.7 LieAlgebras 76
4 JacobiPolynomials 80
4.1 Orthogonality 80
4.2 DifferentialandRecursionFormulas 82
4.3 GeneratingFunctions 88
4.4 FunctionsoftheSecondKind 93
v
vi Contents
4.5 UltrasphericalPolynomials 94
4.6 LaguerreandHermitePolynomials 98
4.7 MultilinearGeneratingFunctions 106
4.8 AsymptoticsandExpansions 115
4.9 RelativeExtremaofClassicalPolynomials 121
4.10 TheBesselPolynomials 123
5 SomeInverseProblems 133
5.1 UltrasphericalPolynomials 133
5.2 BirthandDeathProcesses 136
5.3 TheHadamardIntegral 141
5.4 PollaczekPolynomials 147
5.5 AGeneralization 151
5.6 AssociatedLaguerreandHermitePolynomials 158
5.7 AssociatedJacobiPolynomials 162
5.8 TheJ-MatrixMethod 168
5.9 TheMeixner–PollaczekPolynomials 171
6 DiscreteOrthogonalPolynomials 174
6.1 MeixnerPolynomials 174
6.2 Hahn,DualHahn,andKrawtchoukPolynomials 177
6.3 DifferenceEquations 186
6.4 DiscreteDiscriminants 190
6.5 LommelPolynomials 194
6.6 AnInverseOperator 199
7 ZerosandInequalities 203
7.1 ATheoremofMarkov 203
7.2 ChainSequences 205
7.3 TheHellmann–FeynmanTheorem 211
7.4 ExtremeZerosofOrthogonalPolynomials 219
7.5 ConcludingRemarks 221
8 PolynomialsOrthogonalontheUnitCircle 222
8.1 ElementaryProperties 222
8.2 RecurrenceRelations 225
8.3 DifferentialEquations 231
8.4 FunctionalEquationsandZeros 240
8.5 LimitTheorems 244
8.6 ModificationsofMeasures 246
9 Linearization,ConnectionsandIntegralRepresentations 253
9.1 ConnectionCoefficients 255
9.2 TheUltrasphericalPolynomialsandWatson’sTheorem 261
9.3 LinearizationandPowerSeriesCoefficients 263
9.4 LinearizationofProductsandEnumeration 268
9.5 RepresentationsforJacobiPolynomials 273
9.6 AdditionandProductFormulas 276
9.7 TheAskey–GasperInequality 280
Contents vii
10 TheShefferClassification 282
10.1 Preliminaries 282
10.2 DeltaOperators 285
10.3 AlgebraicTheory 287
11 q-SeriesPreliminaries 293
11.1 Introduction 293
11.2 OrthogonalPolynomials 293
11.3 TheBootstrapMethod 294
11.4 q-Differences 296
12 q-SummationTheorems 299
12.1 BasicDefinitions 299
12.2 ExpansionTheorems 302
12.3 BilateralSeries 307
12.4 Transformations 310
12.5 AdditionalTransformations 313
12.6 ThetaFunctions 315
13 Someq-OrthogonalPolynomials 318
13.1 q-HermitePolynomials 319
13.2 q-UltrasphericalPolynomials 326
13.3 LinearizationandConnectionCoefficients 330
13.4 Asymptotics 334
13.5 Application: TheRogers–RamanujanIdentities 335
13.6 RelatedOrthogonalPolynomials 340
13.7 ThreeSystemsofq-OrthogonalPolynomials 344
14 Exponentialandq-BesselFunctions 351
14.1 Definitions 351
14.2 GeneratingFunctions 356
14.3 AdditionFormulas 358
14.4 q-AnaloguesofLommelandBesselPolynomials 359
14.5 AClassofOrthogonalFunctions 363
14.6 AnOperatorCalculus 365
14.7 Polynomialsofq-BinomialType 371
14.8 Anotherq-UmbralCalculus 375
15 TheAskey–WilsonPolynomials 377
15.1 TheAl-Salam–ChiharaPolynomials 377
15.2 TheAskey–WilsonPolynomials 381
15.3 Remarks 386
15.4 Asymptotics 388
15.5 Continuousq-JacobiPolynomialsandDiscriminants 390
15.6 q-RacahPolynomials 395
15.7 q-IntegralRepresentations 399
15.8 LinearandMultilinearGeneratingFunctions 404
15.9 Associatedq-UltrasphericalPolynomials 410
15.10 TwoSystemsofOrthogonalPolynomials 415
viii Contents
16 TheAskey–WilsonOperators 425
16.1 BasicResults 425
16.2 Aq-Sturm–LiouvilleOperator 432
16.3 TheAskey–WilsonPolynomials 436
16.4 ConnectionCoefficients 442
16.5 BetheAnsatzEquationsofXXZModel 445
17 q-HermitePolynomialsontheUnitCircle 454
17.1 TheRogers–Szego˝ Polynomials 454
17.2 Generalizations 459
17.3 q-DifferenceEquations 463
18 Discreteq-OrthogonalPolynomials 468
18.1 DiscreteSturm–LiouvilleProblems 468
18.2 TheAl-Salam–CarlitzPolynomials 469
18.3 TheAl-Salam–CarlitzMomentProblem 475
18.4 q-JacobiPolynomials 476
18.5 q-HahnPolynomials 483
18.6 q-DifferencesandQuantizedDiscriminants 485
18.7 AFamilyofBiorthogonalRationalFunctions 487
19 Fractionalandq-FractionalCalculus 490
19.1 TheRiemann–LiouvilleOperators 490
19.2 BilinearFormulas 494
19.3 Examples 495
19.4 q-FractionalCalculus 500
19.5 SomeIntegralOperators 503
20 PolynomialSolutionstoFunctionalEquations 508
20.1 Bochner’sTheorem 508
20.2 Differenceandq-DifferenceEquations 513
20.3 EquationsintheAskey–WilsonOperators 515
20.4 LeonardPairsandtheq-RacahPolynomials 517
20.5 CharacterizationTheorems 524
21 SomeIndeterminateMomentProblems 529
21.1 TheHamburgerMomentProblem 529
21.2 ASystemofOrthogonalPolynomials 533
21.3 GeneratingFunctions 536
21.4 TheNevanlinnaMatrix 541
21.5 SomeOrthogonalityMeasures 543
21.6 LadderOperators 546
21.7 Zeros 549
21.8 Theq-LaguerreMomentProblem 552
21.9 OtherIndeterminateMomentProblems 562
21.10 SomeBiorthogonalRationalFunctions 571
22 TheRiemann–HilbertProblem 577
22.1 TheCauchyTransform 577