A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials

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A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials

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Title:
A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials
Creator:
Gishe, Jemal Emina
Place of Publication:
[Tampa, Fla]
Publisher:
University of South Florida
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English

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Subjects / Keywords:
Continuous q-Jacobi polynomials
Lowering operator
Generating function
Weight function
Rodrigues formula
Discriminant
Dissertations, Academic -- Mathematics -- Doctoral -- USF
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous q-Jacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous q-Jacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous q-Jacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,three-term recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2006.
Bibliography:
Includes bibliographical references.
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Title from PDF of title page.
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Document formatted into pages; contains 63 pages.
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Includes vita.
Statement of Responsibility:
by Jemal Emina Gishe.

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A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials
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Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous q-Jacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous q-Jacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous q-Jacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,three-term recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.
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Dissertation (Ph.D.)--University of South Florida, 2006.
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Continuous q-Jacobi polynomials.
Lowering operator.
Generating function.
Weight function.
Rodrigues formula.
Discriminant.
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and ResultantsofChebyshevPolynomials by JemalEminaGishe Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:MouradE.H.Ismail,Ph.D. BorisShekhtman,Ph.D. MasahikoSaito,Ph.D. BrianCurtin,Ph.D. DateofApproval: July13,2006 Keywords:continuousq-Jacobipolynomials,loweringoperator,generatingfunction,weightfunction,Rodriguesformula,discriminant. cCopyright2006,JemalEminaGishe

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IwanttothankDr.MasahikoSaito,Dr.BorisShekhtmanandDr.BrianCurtinmysupervisingcommitteeandDr.DavidRabsonChairpersonofmydefense,whoreadthemanuscriptwithvaluablecomments.IamgratefultoallassistanceIob-tainedfromtheMathematicsdepartmentatUniversityofSouthFlorida;nancially,academicallyandemotionally. Thereareamplewonderfulfriendsinthecourseofmylifewhoinuencedandbelievedinme.JemalDubie,AbduroKelu,A.Hebo,HussienHamda,JimTremmelandMuratThuranareamongthefewtomention. Mywholelifeishighlyindebtedtothesupportandloveofmyfamily.MydreamisrealizedwiththeencouragementandgreatsupportofmyparentsEminaGisheandSinbaChawicha.Theircourage,sacriceandprayertobroughtmeup,inuencedmeandaresourceofmyinspirations.Itisalsoablessingtohaveanintelligent,fullofwondersandcaringbrotherlikeAhmed.Hisdeterminationandstrengthtocopeupwithdiculties,fearlesstoghtinjusticeagainsthimselfandothers,giftednaturetomakefunarefewofhisqualitiestomention. Finally,loveandemotionalsupportIobtainfrommywonderfulwifeZebenayM.Kedirissovaluabletoreachthislevel.HercourageandstrengthtoproperlycareforourprecioussonAnatoliinmyabsenceforyearsandhervisionsoflifewhichhelpedmestaythecourseareafewofherblessedworkthatIcannotaordnottomention.

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1Introduction1 1.1Backgroundandmotivation.......................1 1.2Basics,denitionsandnotation.....................7 1.3Generalpropertiesoforthogonalpolynomials..............10 2AFiniteFamilyofq-OrthogonalPolynomials15 2.1Continuousq-Jacobipolynomials....................16 2.2ThepolynomialsQn...........................25 2.3TheLoweringoperator..........................27 2.4Discriminants...............................38 3ResultantsofChebyshevPolynomials41 3.1Preliminaries...............................41 3.2Chebyshevpolynomialsofsecondkind.................45 3.3Chebyshevpolynomialsofrstkind...................53 References62 AbouttheAuthorEndPagei

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and ResultantsofChebyshevPolynomials JemalEminaGisheAbstract ThesecondproblemisrelatedtorecentresultsofDilcherandStolarky[10]onresultantsofChebyshevpolynomials.TheyusedalgebraicmethodstoevaluatetheresultantoftwocombinationsofChebyshevpolynomialsofthesecondkind.ThisworkprovidesanalternativemethodofcomputingthesameresultantandalsoenablesonetocomputeresultantsofmoregeneralcombinationsofChebyshevpolynomialsofthesecondkind.ResultantsrelatedtocombinationsofChebyshevpolynomialsoftherstkindarealsoconsidered.ii

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Thestudyoforthogonalpolynomialsandspecialfunctionsisanoldbranchofmathematics.Butthebeginningofstudyoforthogonalpolynomialsasadisciplinecanbedatedbackto1894whenStieltjespublishedapaperaboutmomentprobleminrelationtocontinuedfraction.Stieltjesconsideredaboundednon-decreasingfunction(x)intheinterval[0;1)suchthatitsmomentsgivenbyR10xnd(x),forn=0;1;2;:::hasapriorgivensetofvaluesfngasfollows,Z10xnd(x)=n: SimilarresultswhichprecededtheworkofStelieltjesarethoseofChebyshevin1855,whichdiscussedintegralsoftypeR1p(y)

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Manymathematiciansmadeimportantcontributionstothiseldofmathematics.Tomentionafew,Euler'sgammaandbetafunctions;Besselfunctions;PolynomialsofLegenderre,Jacobi,LaguerreandHermite.Mostofthesefunctionswereintro-ducedtosolvespecicproblems.Forexample,Euler'sgammaandbetafunctionsarediscoveredbyEulerinthelate1720's,intheprocessoflookingforafunctionofcontinuousvariablexthatequalsn!whenx=nforanintegern.BesselfunctionsJ(x)=1Xn=0(1)n(x=2)+2n dx2+1 dx+(12

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Inthe1970'sthestudyoforthogonalpolynomialwastakentoanewlevelwithRichardAskey'sleadershipintheareaofspecialfunctionswhileatthesametimeGeorgeAndrewwasadvancingq-seriesandtheirapplicationstonumbertheoryandcombinatorics.TheseadvancementisduetostrongteamworkofMouradIsmail,M.Rahman,G.Gasperamongthefewtomention. Orthogonalpolynomialsandspecialfunctionshaveavarietyofapplicationsinmanyareas.Oneareaisinsolvingdierentialequations.Thesystemsofclassi-calorthogonalpolynomials(suchasJacobi,Hermite,Laguerre)satisfysecondorderdierentialequations.Forexample,Poissonfoundthatthethetafunctionu(x;t)=1Xen2t+inx @x2=@u @t;

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Thebeginningofq-polynomialsisrelatedtotheworkofRogersandRamanujanofthelate19thandearly20thcentury.InthisregardtheRogers-Ramanujanidentities(fornotationreferto(1.2)),1Xn=0qn2 (q;q4;q5)1 (q2;q3;q5)1 (tei;tei;q)1:4

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ThecombinatorialstatementoftheRogers-Ramanujanidentitieswasindepen-dentlydiscoveredbyMacMahonandShurduring1920's.Thecombinatorialinter-pretationofRogers-RamanujanfollowsinthesamewayasEulerinterpretationoftheidentity1Xn=0qn (q;q)1 (q;q)1; Thereisawidevarietyofworkaboutq-seriesinrelationtocombinatoricsandnumbertheory,andtheirapplications,especiallyinrelationtopartitiontheory. Thesystemsofq-orthogonalpolynomialssatisfyasecondorderdierenceequa-tion.HeretheAskey-Wilsonandsomeotherdierenceoperators,usuallycalledraisingandloweringoperators,playtheroleofdierentialoperator.Denitionsoftheseoperators,aregiveninChapter2. ThesecondpartofthedissertationisaboutresultantsrelatedtoChebyshevpoly-nomials.Thetheoryofresultantisanoldandmuchstudiedtopicinwhatusedtobecalledthetheoryofequations[9].DicksonintroducedresultantinhisbookNewFirstCourseintheTheoryofEquation[9]publishedin1939,byconsideringtwofunctionsf(x)=amxm++a0andg(x)=bnxn++b0wheream6=0and5

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Fromtheaboveparagraphweobservethataresultantisascalarfunctionoftwopolynomialswhichisnonzeroifandonlyifthepolynomialsarerelativelyprime.Theresultantoftwopolynomialsisingeneralacomplicatedfunctionoftheircoe-cient.Butthereisanexceptionallyelegantformulaforresultantoftwocyclotomicpolynomialsn(x)(theuniquemonicpolynomialwhoserootsaretheprimitiventhrootofunity).Ithasdegree(n)andwrittenasn(x)=nYk=1;(k;n)=1(xe2ik n); Nowwebrieyintroducethecorecontentofthedissertationandthemethodsusedtosolvetheproblems.Inchapter2,wefollowthestandardnotationbyIsmailasin[14].Insections2:1and2:2,webrieyreviewtheconstructionofcontinuousq-Jacobi6

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Insection2:3,weintroducetheassociatedloweringoperatorandapplythisopera-tortothenormalizedsystemofpolynomials.Here,wedenearelatedscalarproductandshowthatthenormalizedsystemofpolynomialsareeigenfunctionsunderthisscalarproduct.Thiswillleadustoshowtheorthogonalityofnormalizedsystemofpolynomials.Asabyproduct,wewillcomputeaclosedform,three-termrecur-rencerelation,aneigenvalueequation,Rodriguesformulaandageneratingfunctionofthenormalizedcontinuousq-Jacobipolynomials.Asclassicalorthogonalpolyno-mialssatisfysecond-orderdierentialequations,thissystemofpolynomialssatisesasecond-orderdierenceequationswheretheloweringandraisingoperatorstobedenedwillplaytheroleofdierentialoperatorinthelatercase.Inthelastsectionofthischapterwecomputediscriminant,usinganeleganttechniqueintroducedbyIsmail,relatedtothenormalizedpolynomials. Inthelastchapter,wecomputeresultantsofcombinationsofdierentformsrelatedtoChebyshevpolynomialsofrstandsecondkind.Therstsectionofthischapterdealswiththepreliminaries.InthesecondsectionwestateandprovidedierentprooffortheresultantsoftwocombinationsofChebyshevpolynomialsofsecondkindduetoK.DilcherandK.B.Stolarsky[10]andgeneralizetheirresult.ThelastsectionofthischapterdealswiththecorrespondingresultsforChebyshevpolynomialsofrstkind.1.2Basics,denitionsandnotation

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Theq-shiftedfactorialsaredenedas(a;q)0:=1;(a;q)n:=nYk=1(1aqk1)(1.1) forn=1;2;:::;or1,andthemultipleq-shiftedfactorialsaredenedby(a1;a2;:::;ak;q)n:=kYj=1(aj;q)n:(1.2) Thebasichypergeometricseriesisdenedasrsa1;a2;:::;arb1;b2;:::;bsq;z!=rs(a1;a2;:::;ar;b1;b2;:::;bs;q;z)=1Xn=0(a1;a2;:::;ar;q)n 2)n(s+1r): Thenotionofq-shiftedfactorialsandhypergeometricseriesintroducedaboveareextensionsofshiftedandmultishiftedfactorialsbecauseonecanverifythatlimq!1(qa;q)n Thisimpliesthatlimq!1rsqa1;qa2;;qarqb1;qb2;;qbsq;z(1q)s+1r!=rFsa1;a2;;arb1;b2;;bs(1)s+1rz!;rs+18

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(z;q)1;jzj<1;(1.5) BelowisaLemmathatgivesimportantidentitiesinvolvingq-shiftedfactorialswhichwillbeusedinthecourseofthisdissertation.Lemma1.2.2

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ak;(1.9)(a;q)nk=(a;q)n 2k(k+1)nk;(1.10)(a;q1)n=(1=a;q)n(a)nqn(n1)=2:(1.11) 2(1qn=a)(1qn1=a)(1q=a) (1qnk=a)(1qn1k=a)(1q=a)=(1)kqnk+k(k1) 2(q=a;q)n(11=a)(1q=a)(1q1k=a) (1qnk=a)(1qn1k=a)(1q1k=a): existsforalln0.Existenceheremeansthattheresultingintegralsareniteforallnon-negativeintegersn. Then'sarecalledthemomentswithrespecttotheweightfunctionw(x).Inmostcasestheweightfunctionsaresupportedontheniteinterval.Wementionfew10

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Givenaweightfunctionw(x),thenthmonicorthogonalpolynomials(denotedbyQn(x))areconstructedsothattheysatisfyZRQn(x)xkw(x)dx=0;k=0;:::;n1:(1.13) TheaboveequationisequivalenttoZRQn(x)pk(x)w(x)dx=0;k=0;:::;n1; Weintroducethedeterminantn=01n12n+1::::::nn+12n:(1.15) Theexistenceoforthogonalsystemcanbestatedalsoasfollows.11

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foranypositivepolynomialQ(x).Then,themomentintegralispositivedeniteifandonlyifallitsmomentsarerealandn>0. NextistheFundamentalrecurrenceformulawhichstatesthatallsystemsoforthogonalpolynomialssatisesathree-termrecurrencerelation.Fortheproofreferto[14].Theorem1.3.5 withorthogonalityZRn(x)m(x)d(x)=nn;m;(1.19) thenCn>0andn=C1Cn.Itisalsopossibletonotethatforthemoniccasen=n

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Anotherimportantconsiderationinorthogonalpolynomialsisaboutpropertiesofitszeros.Thezerosofsystemoforthogonalpolynomialsaresimpleandinterlace.BelowisatheoremonChristoel-Darbouxidentitieswhichareusedtoverifyaboutthesimplicityandinterlacingpropertiesofzerosoforthogonalpolynomials.Theorem1.3.7

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TheAskey-Wilsonpolynomialsarebuiltthroughthemethodofattachment,whichinvolvesgeneratingfunctionsandsummationtheoremstogetneworthogonalorbiorthogonalfunctions.Webrieyexplainthemethodofattachmentusingthecon-structionofAl-Salam-Chiharapolynomials. Theorthogonalityrelationofcontinuousq-Ultrasphericalpolynomialscanbewrit-teninanequivalentintegralformofZ0(t1ei;t1ei;t2ei;t2ei;e2i;e2i;q)1 Toconstructpolynomialspn(x;t1;t2jq)orthogonalwithrespecttotheweightfunctionw1(x;t1;t2jq)=(e2i;e2i;q)1

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Withtheabovemotivationconcerningthemethodofattachment,thepolynomialsorthogonalwithrespecttotheweightfunctionwhosetotalmassisgivenbytheAskey-Wilsonq-betaintegral,Z0(e2i;e2i;q)1 Thissystemofpolynomialssatisfythefollowingorthogonalityrelationundertheassumptionthatift1,t2,t3,t4arereal,oroccurincomplexconjugatepairsifcomplex,thenmaxfjt1j;jt2j;jt3j;jt4jg<1.Itisknownthat1 2Z11w(x) wherew(x):=w(x;t1;t2;t3;t4jq)=(e2i;q)1 (t1t2;t1t3;t1t4;;q)n;17

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(1t1t2t3t4q2n2)(1t1t2t3t4q2n1): (q1=2q1=2)(z1=z)=2;(2.5) wherex=(z+1=z)=2andf(z)=f((z+1=z)=2):Herex=cosandonecanobservethatz=ei. TheexpansionformulaisanimportantnotionrelatedtotheAskey-Wilsonop-eratorDq.TheAskey-WilsonoperatorDqactsnicelyon(aei;aei;q)n.The(aei;aei;q)nareviewedasthebasisforq-polynomialswhichisanaloguetothebasisxnforclassicalpolynomials.Indeed,forapolynomialsofdegreentheexpan-sionformulaisf(x)=nXk=0fk(aei;aei;q)k;18

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andxk=1 2aqk=2+qk=2 2+1 4,t2=q1 2+3 4,t3=q1 2+1 4,t4=q1 2+3 4inthedenitionoftheAskey-Wilsonpolynomialsafterre-normalizingitfollowsthat,forn=1;2;:::,P(;)n(xjq)=(q+1;q)n 2+1 4ei;q1 2+1 4eiq+1;q1 2(++1);q1 2(++2)q;q!;(2.6) wherex=cos. Theabovepolynomialsarecalledcontinuousq-Jacobipolynomialsandsolvethenormalizedrecurrencerelationxpn(x)=pn+1(x)+1 2[q1 2+1 4+q1 21 4(An+Cn)]pn(x)+1 4An1Cnpn1(x);(2.7) whereP(;)n(xjq)=2nq(1 2+1 4)n(qn+++1;q)n 2++1);q(1 2++2);q)npn(x);(2.8) 2(++1))(1+qn+1 2(++2)) 2+1 4(1q2n+++1)(1q2n+++2);(2.9) andCn=q1 2+1 4(1qn)(1qn+)(1+qn+1 2(+))(1+qn+1 2(++1)) (1q2n++)(1q2n+++1):(2.10) For1 2and1 2,thecontinuousq-Jacobipolynomialshavetheorthogonalityformula19

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2Z11w(x) 2(++2);q1 2(++3);q)1 2(++1);q1 2(++2);q)1(1q++1)(q+1;q+1;q1 2(++3);q)n 2(++1);q)nq(+1 2)nmn; 2+1 4ei;q1 2+3 4ei;q1 2+1 4ei;q1 2+3 4ei;q)12: Forq>1,thecontinuousq-JacobipolynomialsfP(;)n(xjq)gareorthogonalontheimaginaryaxis.Itispossibletorenormalizeinordertoformasystemofpolynomialsorthogonalontherealline. FirstwewilllookatJacobiandq-Hermitepolynomials,forwhichsimilarresultshavealreadybeenconsideredbyM.Ismail.Example2.1.1 (++2): TheclosedformofJacobipolynomialshashypergeometricrepresentationP(;)n(x)=(+1)n

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Usingthechangeofparameters=a+bi,=abi,x!ix,appliedtotherecurrencerelationofJacobipolynomialsandredeningpn(ix)

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(tei;tei;q)1; gives Withtheabovemotivationwenowlookbacktocontinuousq-Jacobipolynomialsforq>1.Forq>1,thesystemofpolynomialsfP(;)n(x)gareorthogonalontheimaginaryaxisandweneedtorenormalizeinordertomakethesepolynomialsorthogonalontherealline.AfterdierenttrialsandrecallingLemma2.0.10,the22

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Thepropernormalizationistomakethefollowingchangeofparametersi:q1 2+1 4:=A=a+bi;(2.11)ii:q1 2+1 4:=B=abi(2.12) andreplacexbyix.Withthisthethree-termrecurrencerelation(2.7)becomesixpn(ix)=pn+1(ix)+iC1pn(ix)+C2pn1(ix):(2.13) TheMapleoutputshowsthatthevaluesofC1andC2arerealconstantsandisin-cludedbelow.Dividing(2.13)byin+1wehavexpn(ix)

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(1q2n2(a2+b2)2)(1q2n1(a2+b2)2)2(1q2n(a2+b2))(1qn)(1+qn(a2+b2)[(1qn1=2a2)2+(1+qn1=2b2)2+2q2n1a2b21]:24

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whereix=cosand=

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(1s1s2s3s4p2n2)(1s1s2s3s4p2n1);(2.17) andBn=s1+s11s11An4Yj=2(1s1sjpn)s1Cn Nowreplacepby1=qandsjbyi=tj,1j4,respectivelyin(2.17)and(2.18)anddenotethetransformedAn,BnandCnbyA0n,B0nandC0n.Thus,A0n=(t1t2t3t4q3n)(1t1t2t3t4qn1) (1t1t2t3t4q2n1)(1t1t2t3t4q2n);C0n=q33n(1qn)Q1j
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Herethepositivityconditionoftherecurrencerelationissuchthat,A00n1C00n>0:(2.24) Itfollowsthatfrom(2.19)and(2.23)thepositivityconditionholdsfor1t1t2t3t4qn2<0: Theaboveinequalityholdsonlyfornitelymanyn,andhencethisnormalizationgivesnitelymanypolynomialsassuggestedbyRouthinLemma2.0.10.Forpurposeofthepositivityconditionofweightfunctionconsideredlaterinthissection,thepairst1andt3,t2andt4areassumedtobecomplexconjugates.2.3TheLoweringoperator (q1=2q1=2)[(z+z1)=2];(2.26)(Af)(x)=1 2[^f(q1=2z)+^f(q1=2z)];(2.27) with^f(z)=f(zz1

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Weintroduceinnerproductwithrespecttow(x)=1 (u2+1)=2 ^g(u)du=Z10^f(u) ^g(q1=2u) (q1=2u2+q1=2)=2duZ10^f(u) ^g(q1=2u) (q1=2u2+q1=2)=2du: wherex=sinh,formabasisinthevectorspaceofallpolynomialsoverC.Theresultofapplyingtheloweringoperatortofn(xjq)gisgivenbythefollowinglemma.Lemma2.3.2 1q(q1=2ae;q1=2ae;q)n1:(2.33)28

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(q1=2q1=2)cosh=2a(1qn) 1q(q1=2ae;q1=2ae;q)n1:2Theorem2.3.3 (1q)(q1=2t1e;q1=2t1e;q)k1=2qn+1 2(1qn)(1qn1t1t2t3t4) (1q)(qt1t2;qt1t3;qt1t4;q)n1

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2+1 4ei;q1 2+3 4ei;q1 2+1 4ei;q1 2+3 4ei;q)12;(2.35) where1=2and1=2.TointroducetheweightfunctionofthepolynomialsQnmakethechangeofparameters(2.11)and(2.12)toobtainw(xjq)=(e2i;q)1 (iAe;q1=2iAe;iBe;q1=2iBe;q)1: itfollowsthatw(x;tjq)=(e2;e2;q)1 Withthis,weintroducetheweightfunction~w(x;tjq)=w(x;tjq) cosh:

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(1q)[1+3+2x(14)]:(2.38) (q1=2q1=2)[(1+q1e2)Q4j=1(1tje) (1+e2)(q1=2e+q1=2e(1+q1e2)Q4j=1(1+tje) (1+e2)(q1=2e+q1=2e]=2~w(x;tjq) (q1=2q1=2)[q1=2e

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(q1)[2sinh(41)13] whichimpliestheresultofthetheorem.2Theorem2.3.5 ~w(x;tjq)=2q(1n)=2 ~w(x;tjq)=(t1t2;t1t3;t1t4;q)n ~w(x;tjq)[2sinh(1qk4)+(qkt1+t2+t3+t4)+qkt1(t2t3+t2t4+t3t4)+t2t3t4]:

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whichimpliesthefollowingtheorem.Theorem2.3.6 ~w(x;tjq)=4(1qn)(1qn14)

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Proof.Firstweshowthatalleigenvaluesarereal.Tothecontraryconsiderthattheaboveeigenvalueproblemhascomplexeigenvalueanditscorrespondingeigenfunc-tionisy.Then,yisaneigenfunctionwitheigenvalue.Withthis,()ZRy(x)

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cosh;(2.43) ~w(x;tjq)=4(1qn)(1qn14) 2q(1n)=2Dq[~w(x;q1=2tjq)Qn1(x;q1=2tjq)]=(1q)2 2Dnq~w(x;qn=2tjq):35

PAGE 41

2Dnq~w(x;qn=2tjq):(2.45) Ageneratingfunctionforasequenceofpolynomialsfpn(x)gisaseriesoftheform1Xn=0npn(x)zn=P(x;z)(2.46) forsomesuitablemultipliersfng.Ifnandpn(x)in(2.46)areassignedandwecandeterminethesumfunctionP(x;z)asanitesumofproductsofanitenumberofknownspecialfunctionsofoneargument,wesaythegeneratingfunctionisknown.Generatingfunctionsplayanimportantroleinthestudyoforthogonalpolynomials.Forexample,someorthogonalpolynomialsaredenedusinggeneratingfunction.Belowarefewsuchpolynomials. TheLegendrepolynomialsPn(x)aregivenas(12xt+t2)1=2=1Xn=0Pn(x)tn; ToconstructthegeneratingfunctionofthepolynomialsQn(x;tjq),itisimpor-tanttomentionthefollowinghypergeometricidentitiesortransformationformulaswhosedetailsaregivenin[14,Chapter12].36

PAGE 42

dn(de=bc;df=bc;q)n whereabc=defqn1,andaqn;qn=(q=a;q)n(a)nqn(n+1)=2;(2.48)(a;q)nk Nowtakinga=t1e,b=t1e,c=qn1t1t2t3t4,d=t1t2,e=t1t3,f=t1t4andapplying(2.47)wehaveQn(sinh;tjq)=(t1t2;q1ne=t3;q1ne=t4;q)nqn1et3t4n430@qn;t1e;t2e;q1n=t3t4t1t2;q1ne=t3;q1ne=t4q;q1A:(2.50) Applying(2.48)wehaveq1ne=t3;q1ne=t4;qn=t3e;t4e;qn(t3t4)nqn(n1)e2n;(2.51) andfrom(2.49)itfollowsthat(qn;q1n=t3t4;q)k Applying(2.51)and(2.52),(2.50)becomesQn(sinh;tjq) (q;t1t2;t3t4;q)n=nXk=0(t1e;t2e;q)k Thelastequationabovegivesthefollowinggeneratingfunctionforthepolynomi-alsQn(x;tjq).37

PAGE 43

(q;t1t2;t3t4;q)ntn=210@t1e;t2et1t2q;te1A210@t3e;t4et3t4q;te1A:(2.54) beapolynomialofdegreenwithleadingcoecient.ThendiscriminantDofg, isdenedby[Dickson,1939]D(g):=2n2Y1j
PAGE 44

Inordertocomputethediscriminantofthepolynomialsunderconsiderationwerstre-normalizetothemoniccase.Wesubstitute(2.11)and(2.12)into(2.7),(2.9)and(2.10),thenapply(2.8)andlet= (1qn)(1qn+1)qn1(x;tjq);(2.60) wherean=(1qnt1t2)(1qn1t1t2t3t4) (1+qn1t2t3)(1+qnt1t3):(2.61) Theclosedformofqn(x;tjq)isqn(x;tjq)=(t1t2;q)n Wecanevaluateqn(x;tjq)atx1=(t1t11)=2andx2=(t3t13)=2.Indeed,qn(x1;tjq)=(t1t2;q)n

PAGE 45

Equatingtheleadingcoecientgivesa=4t1t3q1n 1q:(2.68) Applying(2.63)in(2.67)wehaveax1=b+c(1qn) (1+qn1t1t2);ax2=b+ct3(1qn) Thetheoremfollowsbysolvingtheabovesystemofequations.2Theorem2.4.3

PAGE 46

InthischapterresultantsofdierentformsoflinearcombinationsofChebyshevpolynomialsareconsidered.TheresultingresultantsareexpressibleintermsofChebyshevpolynomialswhosecoecientsandargumentsarerationalfunctionsofthecoecientsinthelinearcombinations.Resultantoftwotermlinearcombination41

PAGE 47

Wefollowclassicaldenitionofresultants.Alternativedenitionsarealsogiven,suchastheonebyH.U.Gerber[12].Denition3.1.1 Belowisatheoremwhichisequivalenttotheabovedenitionandfollowingareworkingcorollaries.Fordetailsandproofsonecanreferto[2].42

PAGE 48

Corollary3.1.3followsfromTheorem3.1.2andusuallycalledfactorizationformula.Thisistheformulawewillbeusinginthecourseofthediscussion.Corollary3.1.4 Equations(3.3)and(3.4)easilyfollowfrom(3.1). ThefollowingmethodisduetoI.Schur,thesketchoftheproofisincludedandifdetailsareneededonecanreferto[21]underdiscriminantsofclassicalpolynomials.Theorem3.1.5

PAGE 49

Withrn(x)=nQnj=1(xj)thenRes(rn(x);sn1(x))=n1nnYj=1sn1(j)=n1n[nYj=1rn1(j)][nYj=1An(j)]:(3.9)44

PAGE 50

Themotivationforthisapproachcamefromtheworkof[15]onthediscriminantsofgeneralorthogonalpolynomials.Inthenextsection,dierentderivationoftheDilcher-Stolarskyresultsisgiven,seeTheorem3.2.4.Then,generalformofcombi-nationsofresultantsofChebyshevpolynomialsofthreeandmoretermsaregiven.InthelastsectionsimilarresultsareestablishedforChebyshevpolynomialsoftherstkind.3.2Chebyshevpolynomialsofsecondkind (3 2)nP(1=2;1=2)n(x) andusuallydenedasUn(x)=sin(n+1) where,x=cosandUltrasphericalpolynomialsarespecialJacobipolynomialswith=.Lemma3.2.2 2m;n:(3.12)45

PAGE 51

WiththeabovemotivationaboutChebyshevpolynomialsofsecondkind,aresultofDilcherandStolarsky[10]isstatedanddierentshortproofisgiven.Thetechniqueandapproachusedhereenabletogeneralizetheresultof[10]andcomputedierentformsofcombinationsofChebyshevpolynomialsofbothrstandsecondkind.Theorem3.2.4 22n(n1)dn(h;k);(3.13) ApplyingthesetwodenitionstheresultofTheorem3.2.4reducestoshowRes(rn(x);sn1(x))=(1)n(n1) 22n(n1)dn(h;k):46

PAGE 52

wherethecoecientsAn(x)andBn(x)willbeevaluatedbelow.Equation(3.16)togetherwithdenitionsofrn(x),sn(x)and(3.11)impliesthatUn1(x)+hUn2(x)=An(x)(Un1(x)+kUn2(x))+Bn(x)(2xUn1(x)Un2(x)+kUn1(x)): 1+k2+2xk;Bn(x)=kh ForlaterusewerewritetheaboveresultforAn(x)asAn(x)=h kx1+hk Thepolynomialrn(x)hasdegreenwithleadingcoecient2n.Thus,rn(x)=2nnYj=1(xxj):

PAGE 53

SinceUn(x)=(1)nUn(x)whichfollowsfromLemma3.1.6andapplyingdenitionofrn(x)itfollowsthatnYj=1sn1(xj)=nhn Tocompletetheproofitremainstocomputen.HereweapplySchur'sresult,Theorem3.1.5aboutthediscriminantoftheclassicalpolynomialswhichismentionedintheintroductionpart. FrominitialconditionsandrecurrencerelationofChebyshevpolynomialsofsec-ondkinditfollowsthatr0(x)=1,r1(x)=2x+k,and2xr1(x)r2(x)=2x(U1(x)+ku0(x))(U2(x)+kU1(x))=U0(x)+2xkU0(x)kU1(x)=r0(x):

PAGE 54

andf(x):=1+(bx+h)(2x+ax+k);g(x):=1+(ax+k)(2x+ax+k):(3.23)Theorem3.2.5 (2+a)22(n1)(n2)Res(f(x);Un(x;a;k)): anditcanbeeasilyveriedthatAn(x)=1+(bx+h)(2x+ax+k) 1+(ax+k)(2x+ax+k);Bn(x)=(ab)x+kh ForlaterusewerewritetheaboveresultforAn(x)asAn(x)=b a(xc1)(xc2) (xd1)(xd2);(3.26) wherecjanddjforj=1;2arerespectivelyzerosoffandgdenedin(3.23).From(3.22)weobservethatUn(x;a;k)ispolynomialofdegreenwithleadingcoef-cient2n1(2+a)andhencewecanassumethat,Un(x;a;k)=2n1(2+a)nYj=1(xxj;n):(3.27)49

PAGE 55

Wenowcomputen.Frominitialconditionsandthree-termrecurrencerelationofChebyshevpolynomialsofsecondkindandusingrepresentation(3.24)itisclearthatU0(x;a;k)=1andU1(x;a;k)=(2+a)x+k.MoreoverUn(x;a;k)satisesthethree-termrecurrencerelation2xUn(x;a;k)=Un+1(x;a;k)+Un1(x;a;k):(3.29) Itfollowsthatn=nYj=1Un1(xj;n;a;k)=2n(n2)(2+a)nnYj=1(xj;nx1;n1)(xj;nxn1;n1)=21(2+a)Un(x1;n1;a;k)Un(xn1;n1;a;k)=(1)n121(2+a)nYj=1Un2(xj;n1;a;k)=(1)n121(2+a)n1: 221n(2+a)n1:(3.30)50

PAGE 56

Clearly,(b(a+2))nUn(c1;a;k)Un(c2;a;k)=Res(f(x);Un(x;a;k)); Theresultofthetheoremnowfollowsfrom(3.31)and(3.32).2Corollary3.2.6 HenceonecanevaluateinclosedformtheresultantsofpolynomialsoftheformP2j=0cjUnjandP2j=0djUnj.51

PAGE 57

Thisisintuitivelyclearforthefollowingreason.Theleft-handsideof(3.34)isapolynomialofdegreem+n1andbyrepeatedusingof(3.11)itcanbeexpressedasP3mk=0kUn+m1k(x)andthereisnolossofgeneralityinassuming0=1,thatisf(x)=2mxm+:ByequatingcoecientsofvariousUj's,wend3m+1linearequationsinthecoecientsoff,gandh.Thetotalnumberofcoecientsinf,g,andhis2(m+1)+mcoecients.Sinceonecoecienthasalreadybeenspeciedweonlyhave3m+1unknownsand3m+1equations,sotheproblemistractableingeneral.ThecaseofChebyshevpolynomialsofrstkindismoretransparent,seeRemark3.3.6. Letx1;x2;:::;xnbethezerosof~vn(x),thatis~vn(x)=2nnYj=1(xxj):(3.35) Moreoverweletf(x)=2mmYk=1(xfk);g(x)=mYk=1(xgk):(3.36) ThefactthatnYj=1g(xj) (xjfk)=n ~vn(fk); ~vn(fk)~n;(3.37) where~n:=nYj=1~vn1(xj)52

PAGE 58

(1 2)nP(1=2;1=2)n; where,x=cosforn=0;1;2;:::.Lemma3.3.2

PAGE 59

2hx+p Onecanexpresssn1(x)asalinearcombinationofrn1(x)andrn(x).Indeed,sn1(x)=An(x)rn1(x)+Bn(x)rn(x):(3.42) andsincefTn(x)gandfUn(x)gsatisfythesamerecurrencerelationfrom(3.17)itfollowsthatAn(x)=1+h(2x+k) 1+k(2x+k);Bn(x)=kh

PAGE 60

kx1+hk From(3.40)weobservethatrn(x)isapolynomialofdegreenwithleadingcoecient2n1.Letfyjgnj=1bezerosofrn(x).Therefore,rn(x)=2n1nYj=1(xyj):(3.44) Applyingthisin3.42itfollowsthatsn1(yj)=An(yj)rn1(yj): 2h yj(1+k2) 2knYj=1rn1(yj)=nhn 2h 2k=nhn 2h+kTn1(1+hk) 2h 2k+kTn1(1+k2) 2k; ChebyshevpolynomialsofrstkindalsohavethepropertyTnz+z1

PAGE 61

Onecaneasilyverifythatthepolynomialsfrn(x)gsatisfytherecurrencerelation2xrn(x)=rn+1(x)+rn1(x): Thetheoremnowfollowsfrom(3.2)and(3.48).2 WithsimilarargumentbeforeitispossibletoexpressTn1(x;b;h)asalinearcom-binationofTn1(x;a;k)andTn(x;a;k).Indeed,Tn1(x;b;h)=An(x)Tn1(x;a;k)+Bn(x)Tn(x;b;k):(3.50) Applying(3.49)itfollowsthatTn1(x)+(bx+h)Tn2(x)=An(x)fTn1(x)+(bx+h)Tn2(x)g+Bn(x)f2xTn1(x)Tn2(x)+(ax+k)Tn1(x)g:56

PAGE 62

1+(ax+k)(2x+ax+k);(3.51)Bn(x)=(ab)x+kh ThisimpliesAn(x)=b a(xc1)(xc2) (xd1)(xd2);(3.53) wherecianddifori=1;2arerespectivelyzerosofquadraticfunctionsofnumeratoranddenominatorofrightsideof(3.51)orrespectivelyzerosoffandgdenedin(3.23). From(3.49)weobservethatTn(x;a;k)isapolynomialofdegreenwithleadingcoecient2n2(2+a).HencewecanassumethatTn(x;a;k)=2n2(2+a)nYj=1(xyj;n):(3.54) TheevaluationofTn1(x;b;h)atthezerosofTn(x;a;k)isgivenbyTn1(yj;n;b;h)=An(yj;n)Tn1(yj;n;a;k):(3.55) Thistogetherwith(3.53)impliesthatnYj=1Tn1(yj;n;b;h)=bn

PAGE 63

AsintheproofofTheorem3.2.5weapplyT0(x;a;k)=1andT1(x;a;k)=(a+1)x+k,three-termrecurrencerelationofChebyshevpolynomialsofrstkindandinductiontoshowthatthesequenceofpolynomialsfTn(x;a;k)gsatisesthefollowingthree-termrecurrencerelation.2xTn(x;a;k)=Tn+1(x;a;k)+Tn1(x;a;k):(3.58) Therefore,n=nYj=1Tn1(yj;n;a;k)=2n3(2+a)nYj=1(yj;ny1;n1)(yj;nyn1;n1)=21nYj=1Tn(y1;n1;a;k)Tn(yn1;n1;a;k)=(1)n121n1: Theabovediscussionimpliesthefollowingtheorem.Theorem3.3.5

PAGE 64

(1=2)nP(1=2;1=2)n(x);(3.61) andUn(x)=(n+1)! (3=2)nP(1=2;1=2)n(x):(3.62) ItfollowsfromthefollowingorthogonalityrelationssatisedbytheabovementionedpolynomialsthatZ11Tn(x)Tm(x)(1x2)1=2dx=0;Z11Un(x)Um(x)(1x2)1=2dx=0;59

PAGE 65

Theexpansionformula(1x)n [[17],p.262]containstheexpansionsofpowersof1xinChebyshevpolynomialsoftherstandsecondkinds,sinceP(;)n(x)=(1)nP(;)n(x)fromLemma3.1.6.Thetermk=0in(3.63)when==1=2seemstobeindeterminatebutcanbefoundbylimitingproceduretobe1=n!.Thus,(1x)n (1=2)k(n+k)!Tk(x):(3.64)Remark3.3.6 and~w(x)=mXj=0djTnj(x):(3.66) AsperRemark3.2.8,ingeneral,thereexistpolynomialsf,gandhofdegreem,mandm1respectivelysuchthatf(x)~wn1=g(x)~vn1(x)+h(x)~vn(x):(3.67) Inthiscasetheanalysisismadesimplerbyexpandingf,gandhinpowersof1x,applying(3.63)andusingTn(x)Tm(x)=1 2[Tm+n(x)+Tmn(x)](3.68) whichfollowsfromthetrigonometricidentitycoscos=1 2[cos(+)+cos(+)]60

PAGE 66

SinceTn(x)=2n1xn+,welet~vn(x)=2n1nYj=1(xxj);f(x)=2mmYk=1(xfk) andg(x)=mYk=1(xgk): ~vn(fk); ~vn(fk)~n;

PAGE 67

G.E.Andrews,R.AskeyandR.Roy,SpecialFunctions.Cambridge,1999.[2] T.M.Apostol,TheresulatantsofthecyclotomicpolynomialsFm(ax)andFn(bx),Math.Comp.29(1975),16,MR0366801(51:3047).[3] R.Askey,Conitiuousq-Hermitepolynomialswhenq>1,InD.Stanton(Ed.),q-SeriesandPartitions,IMAvolumesinMathematicsandItsApplication(pp.151158),NewYork,Springer-Verlag,1989.[4] R.AskeyandJ.Wilson,SomebasichypergeometricorthogonalpolynomialsthatgeneralizesJacobipolynomials.Amer.Math.Soc.,Providence,1985.[5] S.Barnett,MatricesincontrolTheory,2nded.,Krieger,Malabar,Florida,1984.[6] S.Barnett,PolynomialsandLinearControlSystems,MarcelDekker,NewYork,1983.[7] Y.ChenandM.Ismail,LadderOperatorsandDieretialEquationsforOrthog-onalPolynomials.J.Phys.A30(1997),78177829.[8] T.S.Chihara,AnintroductiontoOrthogonalPolynomials,GordonandBreachSciencePublishers,Inc.,NewYork,1978.[9] L.E.Dickson,NewFirstCourseontheTheoryofEquations,Wiley,NewYork,1939.[10] K.Dilcher,andK.B.Stolarsky,ResulatntsandDiscriminantsofChebyshevandrelatedpolynomials,TransactionoftheAmer.Math.Soc.357,no.3(2004),965981.S0002-9947(04)03687-6.[11] I.M.Gelfand,M.M.Kapranov,andA.V.Zelevinsky,Discriminants,Resula-tants,andMultidimensionalDeterminants,BirkhuserBoston,Boston,1994.62

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H.U.Gerber,Wronskiformulaandresultantoftwopolynomilas,thisMONTHLY,91(1984)644646.[13] M.E.H.Ismail,Anelectrostaticmodelforzerosofgeneralorthogonalpolyno-mials,PacicJ.Math.193(2000),355369.[14] M.Ismail,ClassicalandQuantumOrthogonalPolynomialsinonevariable.Cam-bridge,2005.[15] M.E.H.Ismail,Discriminantsandfunctionsofthesecondkindoforthogonalpolynomials,ResultsinMath.34(1998),132149.[16] R.KoekoekandR.Swarttouw,TheAskey-Schemeofhypergeometricorthogonalpolynomialsanditsq-analogues.reportsoftheFacultyofTechnicalMathematicsandInformation94-05,DelftUniversityofTechnology,Delft,1999.[17] E.D.Rainville,SpecialFunctions.NewYork,1960.[18] D.P.Roberts,DiscriminantsofsomePainlevepolynomials,toappear.[19] E.Routh,Onsomepropertiesofcertainsolutionsofadierentialequationofthesecondorder,Proc.LondonMath.Soc.16(1884),245261.[20] J.ShohatandJ.D.Tamarkin,TheproblemofMoments.revisededition,Amer.Math.Soc.,providence,1950[21] G.Szego,OrthogonalPolynomials,fourthedition,AmericanMathematicalSo-ciety,Providence,RhodeIsland,1975.[22] G.N.Watson,AtreatiseonthetheoryofBesselfunctions,2nded.,TheUni-versityPress,Cambridge,1958.63

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IntheFall2001hewasadmittedtoagraduateprograminmathematicsatUniver-sityofSouthFlorida,TampawhereheworkedundersupervisorProf.MouradIsmail.HisscholarlyinterestsareAnalysis,OrthogonalPolynomialsandSpecialFunctions.


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