arithmetic and harmonic sequences
Adhemar Bultheel and Andreas Lasarow
Report TW 687, February 2018
KU Leuven
Department of Computer Science
Celestijnenlaan 200A – B-3001 Heverlee (Belgium)
arithmetic and harmonic sequences
Adhemar Bultheel and Andreas Lasarow
Report TW 687, February 2018
Department of Computer Science, KU Leuven
Abstract
In this paper we study special systems of orthogonal polynomi-als on the unit circle. More precisely, with a view to the recurrence relations fulfilled by these orthogonal systems, we analyze a link of non-negative arithmetic to harmonic sequences as a main subject. Here, arithmetic sequences appear as coefficients of orthogonal poly-nomials and harmonic sequences as corresponding Szeg˝o parameters.
Keywords : Orthogonal polynomials on the unit circle, recurrence relations of Szeg˝o-type, arithmetic sequences, harmonic sequences
Research
Article
Adhemar
Bultheel
and
Andreas
Lasarow
On
orthogonal
polynomials
related
to
arithmetic
and
harmonic
sequences
Abstract:In this paperwe study special systemsof orthogonal polynomials on theunitcircle.Moreprecisely,withaviewtotherecurrencerelationsfulfilledby theseorthogonalsystems,weanalyzealinkofnon-negativearithmetictoharmonic sequencesasamainsubject.Here,arithmeticsequencesappearascoefficientsof orthogonalpolynomialsandharmonicsequencesascorrespondingSzegőparameters. Keywords:Orthogonalpolynomialsontheunitcircle,recurrencerelationsof Szegő-type,arithmeticsequences,harmonicsequences
MSC2010:Primary42C05;secondary30C15
1
Introduction
Throughoutthepaper,let𝑛beapositiveinteger.Supposethat𝑝isa (complex-valued)polynomialofdegree𝑛whichadmitstherepresentation
𝑝(𝑥) = 𝑎𝑛𝑥𝑛+𝑎𝑛−1𝑥𝑛−1+···+𝑎1𝑥1+𝑎0
withnon-negativerealcoefficients.Inthispaper,especially,weareinterestedin thecasethatthedifferenceofconsecutivecoefficientsispositiveandconstant,i.e.
𝑎𝑘−𝑎𝑘−1 = 𝑑, 𝑘=1,...,𝑛,
forsome(arbitrary,butfixed)positiverealnumber𝑑.Wewilldenoteby𝒫𝑛;artheset
ofallpolynomialsofthistype.Thecoefficientsarerelatedtoarithmeticsequences (hencethe"ar"inthenotation)and,since𝑑>0,wehavethemonotonicity
𝑎𝑛 > 𝑎𝑛−1 > ··· > 𝑎1 > 𝑎0 ≥ 0.
Thereby,theconsiderations belowcan beseenasacontinuation ofthose in[2] and[5]onspecialpolynomialsappearinginorthogonalsystemsontheunitcircle
T:={𝑧∈C: |𝑧|=1}ofthecomplexplaneC.
AdhemarBultheel,DepartementComputerwetenschappen,KULeuven,Belgium;e-mail: adhemar.bultheel@cs.kuleuven.be
AndreasLasarow,FakultätInformatik,MathematikundNaturwissenschaften,HTWK Leipzig,Germany;e-mail:andreas.lasarow@htwk-leipzig.de
Supposethat𝜇isameasurebelongingto ℳ,whereℳstandsforthesetof allfinitemeasuresdefinedonthe𝜎-algebraBTofallBorelsubsetsofT.Wewill calla(finiteorinfinite)sequence(𝜑𝑘)𝑘𝜏=0,whereeach𝜑𝑘 isapolynomialofdegree
notgreaterthan𝑘,anorthonormalpolynomialsystemfor𝜇when
∫︁
T
𝜑𝑠(𝑧)𝜑𝑡(𝑧)𝜇(d𝑧) = 𝛿𝑠𝑡,
where𝛿𝑠𝑡:=1if𝑠=𝑡and𝛿𝑠𝑡:=0if𝑠̸=𝑡(foreachchoiceofindices).Hereand
henceforth,𝜏 isanon-negativeintegeror𝜏=∞(arbitrarilychosen,butfixed). Ifthereexistsanorthonormalpolynomial systemforsome𝜇∈ℳ,thenwe findaspecialonewhichisuniquelydeterminedbytheextraconditionthatthe leadingcoefficientofeach𝜑𝑘 isapositiverealnumber.Wewillcallthis(𝜑𝑘)𝜏𝑘=0
the(upto𝜏)normalizedorthonormalpolynomialsystemfor𝜇.
As an aside, we remarkthat there are explicit descriptions of normalized orthonormalpolynomialsystemsbycertaindeterminantformulasor(equivalent) byusingentriesoftheinverseofToeplitzmatricesgivenbytheFouriercoefficients of themeasure𝜇 (see,e.g.,[7],[6],[3],and[4,Section3.6]). Sincewe aremore interestedintherecurrencerelationsforsuchsystems,weomitherethedetails.
Orthonormalpolynomialsystemsforsome𝜇∈ℳfulfillspecific recurrence relations,wheretheelement𝜑𝑛canbecalculatedbasedon𝜑𝑛−1andviceversa.By
thedegreeoffreedomofthechoiceofsuchorthonormalsystemonecansuccessively choose theelementssothattherelated recursionsonly depend ineachstep on someparameterfromtheunitdiskD:={𝑤∈C: |𝑤|<1}(cf.[7],[6,Sections1.5 and1.7],butbelowwefollow moretheapproachof[3],[4,Section3.6]).
Concerningtherecurrencerelations inquestion, thefollowingterm willbe takencenterstage.Wewillcall (𝜙𝑘)𝜏𝑘=0 asequenceofSzegőpolynomialswhen𝜙0
isaconstantfunctiononCwithavalue𝑝0∈C∖{0}andtheotherpolynomialsin thesequencewithhigherindexare(ineach𝑛-thstep)connectedvia
𝜙𝑛(𝑥) = √︀ 1 1−|𝑒𝑛|2 (︁ 𝑥𝜙𝑛−1(𝑥) +𝑒𝑛𝜙̃︀ [𝑛−1] 𝑛−1 (𝑥) )︁ (1)
withsomeSzegőparameter𝑒𝑛∈D(cf.[3]and[4,Definition3.6.7]).Hereinand
furthermore(withsomenon-negativeinteger𝑚),thenotatioñ︀𝑝[𝑚]standsforthe polynomialwhichisuniquelydeterminedvia
̃︀ 𝑝[𝑚](𝑥) = 𝑥𝑚𝑝 (︁ 1 𝑥 )︁ , 𝑥∈C∖{0}, (2) forapolynomial𝑝ofdegreenotgreaterthan𝑚.Inaddition,asequence(𝜙𝑘)𝜏𝑘=0
ofSzegőpolynomialsiscalledcanonicalwhen𝜙0isaconstantfunctiononCwith
InSection2wewillgivesomeinformationaboutsequencesofSzegőpolynomials whichareknown,butwerecallthembecausethatisusefulforourmainresults. Section3formsthebodyofthispaper,wherewestudytheset𝒫𝑛;arwithaviewto
therecurrencerelationsfulfilledbysequencesofSzegőpolynomials.Thisapproach issimilartotheinvestigationsin[2]and[5]betweenpropertiesofcoefficientsof Szegőpolynomialsandcorresponding propertiesofSzegőparameters.However, asamainresultinSection3,itwillberevealedthatanarithmeticsequenceof coefficientsisrelatedtoaharmonicsequenceofSzegőparameters.
Finally,weremarkthattherestrictionofthecasethatthepolynomialshave non-negativerealcoefficientsisnotessential.However,thecalculationsaresomewhat morelabor-intensiveforthemoregeneralsituation.Wewillgivethedetailslater.
2
Hints
on
sequences
of
Szegő
polynomials
Wegive inthissectionsomenotes onasequence(𝜙𝑘)𝜏𝑘=0 ofSzegőpolynomials
whichareusefulforourmainresults.Evenifwemostlyfixthestatementsbelow onlyasaremark,thereisusuallymorethanonelineneededtoprovethemprecisely (dependingontheknowledge).
Since the𝑘-th element𝜙𝑘 of a sequenceof Szegő polynomialsisof degree
notgreaterthan𝑘(infact,ofexactdegree𝑘;cf.[2,Lemma2.4]),therearesome coefficients𝑎𝑘;0,𝑎𝑘;1,...,𝑎𝑘;𝑘 ∈Csothat 𝜙𝑘(𝑥) = 𝑘 ∑︁ 𝑗=0 𝑎𝑘;𝑗𝑥𝑗. (3)
Especially,weareinterestedinthispaperinthecasethatallcoefficientsof thecorrespondingpolynomialsarenon-negativereal.
Remark2.1. Supposethat𝜙𝑛−1isapolynomialofdegree𝑛−1withnon-negative
realcoefficients.If𝜙𝑛 isthepolynomialgivenby(1)withsomerealnumber𝑒𝑛,
where0≤𝑒𝑛<1,then 𝜙𝑛(𝑥) = 𝑛 ∑︁ 𝑗=0 𝑎𝑛−1;𝑗−1+𝑒𝑛𝑎𝑛−1;𝑛−𝑗−1 √︀ 1−𝑒2𝑛 𝑥𝑗
usingthenotationofthecoefficientsasin(3)for𝜙𝑛−1 andsetting𝑎𝑛−1;−1:=0.
Inparticular,wecansee,thatallcoefficientsofthepolynomial𝜙𝑛arenon-negative
realinthissituationaswell.
Thefollowingexamplecanbeseenastheinitialpointoftheconsiderationsinthis paper.Notethatthestructureoftheparameterswhichappearinthisexample
arecloselyrelated tothatin[6,Examples1.6.3and1.6.4] concerning(infinite) sequencesoforthogonalpolynomialsontheunitcircle.
Example2.1. Let𝜙0betheconstantfunctionwithvalue1andlet
𝑒ℓ:=
1
ℓ+1, ℓ=1,2,3,4.
Then the(canonical) sequence(𝜙𝑘)𝑘4=0 of Szegő polynomialswith 𝜙0 and the
sequenceofSzegőparameters(𝑒ℓ)4ℓ=1 isgivenby
𝜙0(𝑥) = 1, 𝜙1(𝑥) = 1 √ 3(2𝑥+1), 𝜙2(𝑥) = 1 √ 6(3𝑥 2 +2𝑥+1), 𝜙3(𝑥) = 1 √ 10(4𝑥 3 +3𝑥2+2𝑥+1), 𝜙4(𝑥) = 1 √ 15(5𝑥 4 +4𝑥3+3𝑥2+2𝑥+1).
Example2.1suggestsalinkbetweenaharmonicsequenceofSzegőparametersand correspondingarithmeticsequencesofcoefficientsofSzegőpolynomials.Thislink willbestudiedinSection3insomewhatmoredetail.
Notethat(1)isequivalentto 𝑥𝜙𝑛−1(𝑥) = 1 √︀ 1−|𝑒𝑛|2 (︁ 𝜙𝑛(𝑥)−𝑒𝑛𝜙̃︀ [𝑛] 𝑛 (𝑥) )︁ (4)
(cf.[2,Lemma2.4]).Thus,inadditiontoRemark2.1, itfollowsfrom(1)thatthe polynomial𝜙𝑛−1canalsobeexpressedintermsofthecoefficientsofthepolynomial
𝜙𝑛.Usingthenotationofthecoefficientsasin(3)for𝜙𝑛,wegetthefollowing.
Remark2.2. Supposethat𝜙𝑛−1isapolynomialofdegree𝑛−1andthat𝜙𝑛 isthe
polynomialgivenby(1)withnon-negativerealcoefficientsandwithsome𝑒𝑛∈D.
Thenweget𝑎𝑛;𝑛>𝑎𝑛;0≥0and0≤𝑒𝑛<1,where
𝜙𝑛−1(𝑥) = 𝑛−1 ∑︁ 𝑗=0 𝑎𝑛;𝑛𝑎𝑛;𝑗+1−𝑎𝑛;0𝑎𝑛;𝑛−𝑗−1 √︁ 𝑎2𝑛;𝑛−𝑎2𝑛;0 𝑥𝑗 and 𝑒𝑛 = 𝑎𝑛;0 𝑎𝑛;𝑛
(cf.[2,Lemma2.6]).Inparticular,ifweknowthecoefficientsof thepolynomial 𝜙𝑛,thentheparameter𝑒𝑛 andthepolynomial𝜙𝑛−1 areuniquelydetermined.
Thenextexampleillustratesthatthecase𝑛=1concerning(1)andsequencesof Szegőpolynomialsisunspectacular(butanexception).
Example2.2. Supposethat𝑝isapolynomialadmitting𝑝(𝑥)=𝑎1𝑥+𝑎0withsome
non-negativerealnumbers𝑎1 and𝑎0.WithaviewtoRemark2.2onecanseethat
thereisa(𝜙𝑘)1𝑘=0ofSzegőpolynomialswith𝜙1=𝑝ifandonlyif𝑎1>𝑎0,where
theSzegőpolynomialswith𝜙1 =𝑝isthenuniquelydetermined andcanonical
(since𝜙0 istheconstantfunctionwithvalue
√︀
𝑎2 1−𝑎20).
Withthefollowingexamplewewillemphasizethat,undertheexclusivetermsof Remark2.2,itispossiblethatacoefficientofthepolynomial𝜙𝑛−1isnegativereal.
Example2.3. If𝜙2 and𝜙3arethepolynomialsgivenby
𝜙2(𝑥) = 8𝑥2−𝑥+ 7 and 𝜙3(𝑥) = 10𝑥3+ 4𝑥2+ 8𝑥+ 6,
then(1)isfulfilledwith𝑛=3and𝑒𝑛=35.
InviewofthedefinitionofsequencesofSzegőpolynomials,afinitesequenceofthis typecanbeextendedtoaninfiniteone(bychoosingthemissingSzegőparameters arbitrary,butbelongingtoD).Thefollowingnotepointsoutasimpleextension regarding(infinite)orthonormalpolynomialsystemsbyfixingthemeasure𝜇. Remark2.3. Suppose that (𝜙𝑘)𝑘𝑛=0 is a sequence of Szegő polynomials. Then
(𝜙𝑘)𝑛𝑘=0isanorthonormalpolynomialsystemfor𝜇givenby
𝜇(𝐵):= 1 2𝜋 ∫︁ 𝐵 1 |𝜙𝑛(𝑧)|2 𝜆(d𝑧), 𝐵∈BT, (5)
where𝜆standsforthe linearLebesgue-BorelmeasureonT.Infact(cf.[6, Theo-rems1.7.5and 1.7.8]or[2,Proposition2.5andRemark5.2]),ifwechoose
𝜙𝑛+ℓ(𝑥):=𝑥ℓ𝜙𝑛(𝑥), ℓ=1,2,...,
then(𝜙𝑘)∞𝑘=0 isasequenceofSzegőpolynomialswithparameter𝑒𝑛+ℓ=0forall
integersℓ≥1,where(𝜙𝑘)∞𝑘=0isanorthonormalpolynomialsystemfor𝜇.
If(𝜙𝑘)𝑛𝑘=0isasequenceofSzegőpolynomials,thenRemark2.3clarifiesparticularly
thatthereisa𝜇∈ℳsothat(𝜙𝑘)𝑛𝑘=0isanorthonormalpolynomialsystemfor𝜇.
Inadditiontothat(cf.[6,Theorem1.7.11]or[4,Theorem3.6.2]),ifweconsideran infinitesequence(𝜙𝑘)∞𝑘=0ofSzegőpolynomials,thenthere isexactlyonemeasure
𝜇∈ℳsothat(𝜙𝑘)𝑘∞=0 isanorthonormalpolynomialsystemfor𝜇.
Notethat,conversely,ifwehaveanorthonormalpolynomialsystem(𝜑𝑘)𝜏𝑘=0
for𝜇∈ℳ,thenintheset ofall suchsystemsareincluded sequencesof Szegő polynomialsandwefindaspecialonewhichiscanonical(cf.[6,Chapter1]or[4, Section3.6]).
Finally,werecallthefollowingmanipulationbymultiplicationofsequencesof Szegőpolynomials(cf.[2,Remark2.10]).
Remark2.4. Supposethat(𝜙𝑘)𝑘𝜏=0isasequenceofSzegőpolynomialswithrelated
sequence(𝑒ℓ)𝜏ℓ=1ofSzegőparametersandlet𝑎beapositiverealnumber.Then
(𝑎𝜙𝑘)𝜏𝑘=0 is a sequence of Szegő polynomials with sequence (𝑒ℓ)𝜏ℓ=1 of Szegő
parameters.Furthermore,if(𝜙𝑘)𝑘𝜏=0 isanorthonormalpolynomialsystemforthe
measure𝜇,then(𝑎𝜙𝑘)𝜏𝑘=0 isanorthonormalpolynomialsystemfor 𝑎12𝜇.
3
On
Szegő
polynomials
belonging
to
𝒫
𝑛;arNow,withaviewtotheset𝒫𝑛;ar,westudyspecialsequencesofSzegőpolynomials.
Thereby,thefollowingresultisthelynchpin.
Lemma3.1. Suppose that 𝜙𝑛−1 is a polynomial of degree 𝑛−1 and that 𝜙𝑛 is
the polynomial given by (1) with some 𝑒𝑛 ∈ D and 𝑛 ≥2. If 𝜙𝑛 ∈ 𝒫𝑛;ar, then
𝜙𝑛−1∈𝒫𝑛−1;arand, usingthenotationof thecoefficientsasin (3)for𝜙𝑛,then
𝜙𝑛−1(𝑥) = 𝑛−1 ∑︁ 𝑗=0 (𝑗+1)𝑑(𝑛𝑑+ 2𝑎𝑛;0) √︁ 𝑎2 𝑛;𝑛−𝑎2𝑛;0 𝑥𝑗, (6) 𝑒𝑛 = 𝑎𝑛;0 𝑛𝑑+𝑎𝑛;0 (7) with𝑑=𝑎𝑛;1−𝑎𝑛;0>0.
Proof. Let𝜙𝑛∈𝒫𝑛;ar.Withaviewto(3)for𝜙𝑛,weget𝑎𝑛;𝑛>𝑎𝑛;0≥0,
𝜙𝑛−1(𝑥) = 𝑛−1 ∑︁ 𝑗=0 𝑎𝑛;𝑛𝑎𝑛;𝑗+1−𝑎𝑛;0𝑎𝑛;𝑛−𝑗−1 √︁ 𝑎2𝑛;𝑛−𝑎2𝑛;0 𝑥𝑗, (8) and 𝑒𝑛 = 𝑎𝑛;0 𝑎𝑛;𝑛 (9)
(cf.Remark2.2).Furthermore,since𝜙𝑛∈𝒫𝑛;ar,thereisa𝑑>0sothat
𝑎𝑛;𝑘−𝑎𝑛;𝑘−1 = 𝑑, 𝑘=1,...,𝑛.
Thus,itfollowsthat
𝑒𝑛 = 𝑎𝑛;0 𝑎𝑛;𝑛 = 𝑎𝑛;0 𝑛𝑑+𝑎𝑛;0 , i.e.(7),and 𝑎𝑛;𝑛𝑎𝑛;𝑗+1−𝑎𝑛;0𝑎𝑛;𝑛−𝑗−1 = (𝑛𝑑+𝑎𝑛;0) (︀ (𝑗+1)𝑑+𝑎𝑛;0 )︀ −𝑎𝑛;0 (︀ (𝑛−𝑗−1)𝑑+𝑎𝑛;0 )︀ = 𝑛(𝑗+1)𝑑2+𝑛𝑑𝑎𝑛;0+(𝑗+1)𝑑𝑎𝑛;0−(𝑛−𝑗−1)𝑑𝑎𝑛;0 = (𝑗+1)𝑑(𝑛𝑑+2𝑎𝑛;0)
for𝑗=0,1,...,𝑛−1,i.e.weget(6).Inparticular,wesee𝜙𝑛−1∈𝒫𝑛−1;ar.
Theorem3.1. Suppose that 𝑝 ∈ 𝒫𝑛;ar for some 𝑛 ≥ 2. Then there is a uniquely
determined sequence (𝜙𝑘)𝑛𝑘=0 ofSzegőpolynomials,where 𝜙𝑛=𝑝.Thissequence
(𝜙𝑘)𝑛𝑘=0 of Szegőpolynomials is canonical,where the associatedSzegőparameter
𝑒𝑛 is givenvia (7)and (3).Furthermore, thepolynomial𝜙ℓ belongs to𝒫ℓ;ar and
theassociatedSzegőparameter𝑒ℓ isgivenby
𝑒ℓ =
1
ℓ+1, ℓ=1,...,𝑛−1.
Proof. We set 𝜙𝑛 =𝑝 andusethe notation(3)with𝑘 replaced by 𝑛.Because
𝑝∈𝒫𝑛;ar,wehave𝑎𝑛;𝑛>𝑎𝑛;0≥0sothattheparameter𝑒𝑛andthepolynomial
𝜙𝑛−1 accordingto(9)and(8),respectively,arewell-defined.Sincetherelation(4)
isequivalentto(1),from(8)and(9)weget(1).Furthermore,Lemma3.1yields 𝜙𝑛−1∈𝒫𝑛−1;arandtherepresentation(7)and(6)for𝑒𝑛and𝜙𝑛−1,respectively.In
particular,if𝑛−1≥2,wecanproceedwiththeapproachandget𝜙𝑛−2∈𝒫𝑛−2;ar,
whereRemark2.2and(6)imply
𝑒𝑛−1 = 𝑎𝑛−1;0 𝑎𝑛−1;𝑛−1 = 𝑑(𝑛𝑑+2𝑎𝑛;0) √︁ 𝑎2 𝑛;𝑛−𝑎2𝑛;0 𝑛𝑑(𝑛𝑑+2𝑎𝑛;0) √︁ 𝑎2 𝑛;𝑛−𝑎2𝑛;0 = 𝑑(𝑛𝑑+ 2𝑎𝑛;0) 𝑛𝑑(𝑛𝑑+ 2𝑎𝑛;0) = 1 𝑛.
Forℓ=1,...,𝑛−1,bytheprincipleofinduction,weget𝜙ℓ∈𝒫ℓ;arand𝑒ℓ = ℓ+11 .
Finally,withaviewtoExample2.2,onecanseethattheconstructionusedabove leadstoa uniquelydetermined sequence (𝜙𝑘)𝑛𝑘=0 of Szegőpolynomials, where
𝜙𝑛=𝑝,andthatthisiscanonical.
Corollary3.1. If𝑝∈𝒫𝑛;arwith 𝑛≥1, thenallzerosof𝑝belongtoD.
Proof. For𝑛=1,thestatementfollowsimmediatelyfromthedefinitionof𝒫1;ar.
If 𝑛 ≥2, thenTheorem 3.1 implies that there isa sequence(𝜙𝑘)𝑛𝑘=0 of Szegő
polynomialswith𝜙𝑛=𝑝.Thus,inthiscase,thestatementfollowsfromageneral
resultonsequencesofSzegőpolynomials(see,e.g.,[2,Proposition2.5(a)]). Asanaside,wenotethatthestatementofCorollary3.1followsfromaclassical theoremduetoEneström–Kakeyaaswell(see,e.g.,[1,TheoremA]).
Theresult,thatisrevealedbyTheorem3.1,comprisesaveryspecialstructure ofSzegőpolynomials.Thiswillbeemphasizedbythefollowing.
Proposition3.1. Suppose that 𝑝 ∈ 𝒫𝑛;ar for some 𝑛 ≥ 2 and let (𝜙𝑘)𝑛𝑘=0 be the
uniquelydeterminedsequenceofSzegőpolynomials,where𝜙𝑛=𝑝.
(a) The sequence (𝜙𝑘)𝑛𝑘=0 is the (up to 𝑛) normalized orthonormal polynomial
(b) Let𝑑𝑚 bethe differenceof consecutive coefficientsof𝜙𝑚 for𝑚=1,2,...,𝑛
and let𝑑0 := 𝜙0(0). Then 𝑑0 > 𝑑1 >··· >𝑑𝑛−1 ≥ 𝑑𝑛, where 𝑑ℓ = 𝜙ℓ(0)
and𝑑ℓ−1=𝑑ℓ
√︁
1 +2ℓ forℓ=1,...,𝑛−1andwhere 𝑑𝑛−1=𝑑𝑛
√︁
1 +2𝜙𝑛(0)
𝑛𝑑𝑛 .
Furthermore,𝑑𝑚=𝑚1(︀𝜙̃︀𝑚[𝑚](0)−𝜙𝑚(0))︀for𝑚=1,2,...,𝑛.
(c) Denotingthecoefficientsasin (3),then
𝑎𝑘;𝑗 = 1 𝑗+1 √︂ 𝑛+1 𝑘2+3𝑘+2 (︁ (︀ ̃︀ 𝑝[𝑛](0))︀2 −(︀ 𝑝(0))︀2 )︁ , 𝑗=0,1,...,𝑘, forallindices𝑘=0,1,...,𝑛−1.
(d) Thefollowingstatementsareequivalent: (i) 𝑝(0)=0.
(ii) 𝑒𝑛=0.
(iii) 𝑑𝑛=𝑑𝑛−1.
(e) Thefollowingstatementsareequivalent: (iv) 𝑝(0)=𝑑𝑛. (v) 𝑒𝑛=𝑛+11 . (vi) 𝑑𝑛=𝑑𝑛−1 √︁ 𝑛 𝑛+2.
Proof. Asiswell-known(cf.Remark2.3),thesequence(𝜙𝑘)𝑛𝑘=0 isanorthonormal
polynomial system for the measure 𝜇 given by (5). Since 𝜙𝑛 = 𝑝 particularly
implies 𝜙𝑛 ∈𝒫𝑛;ar andsince Theorem3.1 yields 𝜙ℓ ∈𝒫ℓ;ar forℓ=1,...,𝑛−1
aswellasthat𝜙0(0)isapositiverealnumber,theleadingcoefficient of𝜙𝑘 isa
positiverealnumberforeach𝑘=0,1,...,𝑛.Thus,thesequence(𝜙𝑘)𝑛𝑘=0isthe
(upto𝑛)normalizedorthonormalpolynomialsystemfor𝜇.Hence,(a)isproven. Because 𝜙𝑚 ∈ 𝒫𝑚;ar, the number𝑑𝑚 iswell-defined according to(b) and
̃︀
𝜙𝑚[𝑚](0)istheleadingcoefficientof𝜙𝑚 for𝑚=1,2,...,𝑛.Consequently,based
on(3),for𝑚=1,2,...,𝑛wehave
̃︀
𝜙𝑚[𝑚](0) = 𝑎𝑚;𝑚 = 𝑚𝑑𝑚+𝑎𝑚;0 = 𝑚𝑑𝑚+𝜙𝑚(0),
i.e.𝑑𝑚=𝑚1(︀𝜙̃︀
[𝑚]
𝑚 (0)−𝜙𝑚(0))︀.Moreover,byTheorem3.1andLemma3.1follows
therepresentation(6)for𝜙𝑛−1,where𝑑=𝑑𝑛.Thisimplies
𝑑𝑛−1 = 𝑑𝑛(𝑛𝑑𝑛+ 2𝑎𝑛;0) √︁ 𝑎2 𝑛;𝑛−𝑎2𝑛;0 = √︁𝑑𝑛(𝑛𝑑𝑛+ 2𝑎𝑛;0) (𝑛𝑑𝑛+𝑎𝑛;0)2−𝑎2𝑛;0 = √︀𝑑𝑛(𝑛𝑑𝑛+ 2𝑎𝑛;0) (𝑛𝑑𝑛)2+ 2𝑛𝑑𝑛𝑎𝑛;0 = 𝑛𝑑 2 𝑛(1+ 2𝑎𝑛;0 𝑛𝑑𝑛 ) 𝑛𝑑𝑛 √︁ 1+2𝑎𝑛;0 𝑛𝑑𝑛 = 𝑑𝑛 √︂ 1+2𝑎𝑛;0 𝑛𝑑𝑛 = 𝑑𝑛 √︂ 1+2𝜙𝑛(0) 𝑛𝑑𝑛 .
Inparticular,taking𝜙𝑛(0)≥0and𝑑𝑛>0intoaccount, wesee𝑑𝑛−1≥𝑑𝑛.Let
hand𝑒ℓ= 𝑎ℓ;0
ℓ𝑑ℓ+𝑎ℓ;0 byLemma3.1andRemark2.2.Hence,weget𝑎ℓ;0̸=0and
𝑑ℓ = 𝑎ℓ;0 = 𝜙ℓ(0).
Therefore,similarasabove,inviewofTheorem3.1andLemma3.1itfollows
𝑑ℓ−1 = 𝑑ℓ
√︂
1+2
ℓ, ℓ=1,...,𝑛−1, andparticularly𝑑ℓ−1>𝑑ℓ.Hence,(b)isproven.
By(b)and𝑝=𝜙𝑛,wehave(notẽ︀𝑝
[𝑛](0)>𝑝(0)) 𝑑𝑛 √︂ 1+2𝜙𝑛(0) 𝑛𝑑𝑛 = 1 𝑛 (︀ ̃︀ 𝑝[𝑛](0)−𝑝(0))︀ √︃ 1 + 2𝑝(0) ̃︀ 𝑝[𝑛](0)−𝑝(0) = 1 𝑛 (︀ ̃︀ 𝑝[𝑛](0)−𝑝(0))︀ √︃ ̃︀ 𝑝[𝑛](0) +𝑝(0) ̃︀ 𝑝[𝑛](0)−𝑝(0) = 1 𝑛 √︁ (︀ ̃︀ 𝑝[𝑛](0))︀2 −(︀ 𝑝(0))︀2 .
Furthermore,bytheprincipleofinduction,onecanshowthat
𝑛−1 ∏︁ 𝑗=ℓ (︁ 1+2 𝑗 )︁ = 𝑛+𝑛 2 ℓ+ℓ2, ℓ=𝑛−1,...,1.
Thus,usingthecoefficientsasin(3),from(b)weget
𝑎𝑘;0 = 𝜙𝑘(0) = 𝑑𝑘 = √︂ 1 +2𝜙𝑛(0) 𝑛𝑑𝑛 𝑛 ∏︁ ℓ=𝑘+1 𝑑ℓ = √︂ 1 +𝑛 𝑘+ 1 + (𝑘+ 1)2 (︁ (︀ ̃︀ 𝑝[𝑛](0))︀2 −(︀ 𝑝(0))︀2 )︁ = √︂ 𝑛+1 𝑘2+3𝑘+2 (︁ (︀ ̃︀ 𝑝[𝑛](0))︀2 −(︀ 𝑝(0))︀2)︁
for𝑘=0,...,𝑛−1.Recalling𝜙ℓ∈𝒫ℓ;ar and𝑑ℓ=𝜙ℓ(0)forℓ=1,...,𝑛−1,the
assertionof(c)follows.
Taking 𝑝 = 𝜙𝑛 and (7) into account the assertions of (d) and (e) are a
consequenceof(b).
Corollary3.2. Supposethat (𝜙𝑘)𝜏𝑘=0 isasequence ofSzegőpolynomials.
(a) If𝜙ℓ∈𝒫ℓ;ar forsome indexℓwith 𝜏 >ℓ≥1, where thedifferenceof
conse-cutivecoefficientsof thepolynomial𝜙ℓ isnotequal to𝜙ℓ(0),then𝜙𝑘̸∈𝒫𝑘;ar
forallindices𝑘 with𝜏≥𝑘>ℓ.
(b) If 𝜙ℓ ∈ 𝒫ℓ;ar for some index ℓ with 𝜏 > ℓ ≥ 1, where the associated Szegő
Proof. Let 𝜙ℓ ∈ 𝒫ℓ;ar for an index ℓ with 𝜏 > ℓ ≥ 1, where the difference 𝑑ℓ
ofconsecutivecoefficientsof𝜙ℓisdifferentfrom𝜙ℓ(0).Furthermore,weassume
temporarily that there isan index 𝑘 with𝜏 ≥ 𝑘 >ℓ, where 𝜙𝑘 ∈𝒫𝑘;ar.Thus,
Theorem3.1andpart(b)of Proposition3.1with𝑝=𝜙𝑘 yield𝑑ℓ=𝜙ℓ(0).But
thisconflictswiththeconditionof𝜙ℓ.Therefore,𝜙𝑘̸∈𝒫𝑘;arforallindices𝑘with
𝜏 ≥𝑘 >ℓ,and(a) isproven. Sincethecondition 𝑒ℓ =0leadsto𝜙ℓ(0)=0(cf.
Remark2.2),thestatementof(b)isaconsequenceof(a).
InviewoftheinterdependencyofsequencesofSzegőpolynomialsandorthonormal polynomialsystemsformeasures𝜇∈ℳ(see,e.g.,thenotesfromRemark2.3to theendofSection2),wecansimplyreformulatethestatementsaboveintermsof orthonormalsystems.Inparticular,wegetthefollowingresultwhichturnsoutto besomewhatmoresurprising(ifyoudonothavetherecurrencerelationinmind). Theorem3.2. Supposethat(𝜙𝑘)𝑘𝜏=0isanorthonormalpolynomialsystemforsome
measure𝜇∈ℳ.
(a) If𝜙𝑛∈𝒫𝑛;ar forsomeindex𝑛with 𝑛≥2, then𝑎𝑘;0̸=0and
𝑎𝑘;𝑗 = 𝑢𝑘 𝑗+1 √︂ 𝑛+1 𝑘2+3𝑘+2 (︀ 𝑎2𝑛;𝑛−𝑎2𝑛;0 )︀ , 𝑗=0,1,...,𝑘, withsome𝑢𝑘 ∈Tforallindices𝑘=0,1,...,𝑛−1basedon (3).
(b) If𝜙ℓ∈𝒫ℓ;ar for some indexℓwith 𝜏 >ℓ≥1, wherethe differenceof
conse-cutivecoefficientsof thepolynomial𝜙ℓ isnotequalto𝜙ℓ(0),then𝜙𝑘 ̸∈𝒫𝑘;ar
forallindices𝑘with𝜏 ≥𝑘>ℓ.
(c) If 𝜙ℓ ∈ 𝒫ℓ;ar for some index ℓ with 𝜏 > ℓ ≥ 1, where 𝜙ℓ(0) is zero, then
𝜙𝑘̸∈𝒫𝑘;ar forallindices𝑘with𝜏 ≥𝑘>ℓ.
BasedonTheorem3.1,wecanalsoseethatthecaseofaninfinitesequenceofSzegő polynomials,wherethecoefficientsofallpolynomialsarerelatedtonon-negative realarithmeticsequences,isaveryspecialone.
Theorem3.3. Suppose that (𝜙𝑘)𝑘∞=0 is a sequence of Szegő polynomials and let
(𝑒ℓ)∞ℓ=1 be theassociated sequence ofSzegőparameters. Then thefollowing
state-mentsareequivalent:
(i) Thereisapositivereal number𝑝0 sothatthesequence(𝜙𝑘)∞𝑘=0 isgivenby
𝜙𝑘(𝑥) = 𝑝0 √ 𝑘2+3𝑘+2 𝑘 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗, 𝑘=0,1,2,3,....
(ii) Foreachindexℓwithℓ≥1thepolynomial𝜙ℓ belongs to𝒫ℓ;ar and𝜙0(0)isa
positiverealnumber.
(iii)Thereissome ℓ0≥1sothat foreach indexℓwith ℓ≥ℓ0 the polynomial𝜙ℓ
(iv)Thesequence (𝑒ℓ)∞ℓ=1 isgiven by
𝑒ℓ =
1
ℓ+1, ℓ=1,2,3,..., and𝜙0(0)isapositivereal number.
Inparticular, if (i)is fulfilled,then(𝜙𝑘)∞𝑘=0 isthe normalized orthonormal
poly-nomialsystemforthe(uniquelydetermined)measure𝜇 givenby 𝜇(𝐵):= 1
𝑝20𝜋
∫︁
𝐵
(1−ℜe𝑧)𝜆(d𝑧), 𝐵∈BT, (10) where𝜆standsforthelinearLebesgue-BorelmeasureonT.
Proof. The implications “(i)⇒(ii)” and“(ii)⇒ (iii)” area consequenceof the settings.Furthermore,theimplications“(ii)⇒(iv)” and“(iii)⇒(ii)” followfrom Theorem3.1.Itremainstoprovethat“(iv)⇒(i)”.Therefore,wesupposethat(iv) holds.Since(𝜙𝑘)∞𝑘=0isasequenceofSzegőpolynomials,foreachpositiveinteger
𝑛,therelation (1)isfulfilledwiththespecialSzegőparameter𝑒𝑛 givenby(iv).
Takingintoaccountthat𝜙0(0)isapositiverealnumber,i.e.𝜙0 istheconstant
functionwith(positive) value𝜙0(0),wehave
𝜙0(𝑥) = 𝜙0(0) = 𝑝0 √ 02+3·0+2 0 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗
withthepositiverealnumber𝑝0:=
√
2𝜙0(0)and(1)for𝑛=1implies
𝜙1(𝑥) = 1 √︁ 1−(︀1 2 )︀2 (︁ 𝑥𝑝0+ 1 2𝑝0 )︁ = √ 𝑝0 12+3·1+2 1 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗.
Now,bytheprincipleofinduction,wesupposethat𝜙𝑘 isgivenby
𝜙𝑘(𝑥) = 𝑝0 √ 𝑘2+3𝑘+2 𝑘 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗,
forsomepositiveinteger𝑘.Thenwehave
̃︀ 𝜙𝑘[𝑘](𝑥) = √ 𝑝0 𝑘2+3𝑘+2 𝑘 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑘−𝑗, where √︂ 1−(︁ 1 𝑘+2 )︁2√︀ 𝑘2+3𝑘+2 = 1 𝑘+2 √︁ (︀ (𝑘+2)2−1)︀ (𝑘2+3𝑘+2) = 1 𝑘+2 √︀ (𝑘+1)(𝑘+3)(𝑘+1)(𝑘+2) = (𝑘+1) √︂ 𝑘+3 𝑘+2,
𝑗+𝑘−𝑗+1 𝑘+2 = 𝑗𝑘+2𝑗+𝑘−𝑗+1 𝑘+2 = (𝑘+1)(𝑗+1) 𝑘+2 , and (𝑘+2)(𝑘+3) = 𝑘2+5𝑘+6 = (𝑘+1)2+3(𝑘+1)+2. Thus,inviewof(iv)and(1)for𝑛=𝑘+1,weget
𝜙𝑘+1(𝑥) = 1 √︁ 1−(︀ 1 𝑘+2 )︀2 ·√ 𝑝0 𝑘2+3𝑘+2· (︁ 𝑥 𝑘 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗+ 1 𝑘+2 𝑘 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑘−𝑗 )︁ = 𝑝0 𝑘+1· √︂ 𝑘+2 𝑘+3· (︁𝑘+1 𝑘+2𝑥 0 + 𝑘 ∑︁ 𝑗=1 (︁ 𝑗+𝑘−𝑗+1 𝑘+2 )︁ 𝑥𝑗+ (𝑘+1)𝑥𝑘+1)︁ = √︀ 𝑝0 (𝑘+2)(𝑘+3) 𝑘+1 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗 = √︀ 𝑝0 (𝑘+1)2+3(𝑘+1)+2 𝑘+1 ∑︁ 𝑗=0 (𝑗+1)𝑥𝑗,
sothatwehaveproven,bytheprincipleofinduction,that(i)followsfrom(iv). Supposethat(i)holds.If𝑝0=
√
2,thentheconsiderationsin[6,Example1.6.4] imply that (𝜙𝑘)∞𝑘=0 isthe normalized orthonormal polynomial system for the
(uniquelydetermined)measure 𝜇given by(10).Using thisspecialcase in com-bination withRemark2.4, we seethat thisholds for anypositive realnumber 𝑝0.
Finally,inadditiontoLemma3.1andCorollary3.2,wepresentthefollowingresult concerningtheone-stepextensiongivenby(1).
Proposition3.2. Supposethat𝜙𝑛−1∈𝒫𝑛−1;arforsome𝑛≥2andlet𝑑𝑛−1 bethe
differenceofconsecutive coefficientsofthepolynomial𝜙𝑛−1.Furthermore, let𝜙𝑛
bethepolynomialgivenby (1)andsomeparameter 𝑒𝑛∈D, where
𝜙𝑛(𝑥) = 𝑎𝑛;𝑛𝑥𝑛+𝑎𝑛;𝑛−1𝑥𝑛−1+···+𝑎𝑛;1𝑥1+𝑎𝑛;0
withsome coefficients𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛∈Casin (3).
(a) Thefollowingstatementsareequivalent:
(i) Theparameter𝑒𝑛 isarealnumberwith0≤𝑒𝑛<1and𝜙𝑛−1(0)=𝑑𝑛−1.
(ii) The polynomial𝜙𝑛 belongsto𝒫𝑛;ar.
Inparticular,if(i)issatisfied,thenthecoefficients𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛 ofthe
polynomial𝜙𝑛 arereal and
𝑎𝑛;𝑛 > 𝑎𝑛;𝑛−1 > ··· > 𝑎𝑛;1 > 𝑎𝑛;0 ≥ 0. (11)
(b) Supposethat𝜙𝑛−1(0)̸=0,butalso that𝜙𝑛−1(0)̸=𝑑𝑛−1, andlet
𝑚:=min {︁𝜙 𝑛−1(0) 𝑑𝑛−1 , 𝑑𝑛−1 𝜙𝑛−1(0) }︁ . Thenthefollowingstatementsareequivalent:
(iii)Theparameter𝑒𝑛 isa realnumberwith0≤𝑒𝑛≤𝑚.
(iv)Thecoefficients𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛 ofthepolynomial𝜙𝑛 arereal and
𝑎𝑛;𝑛 ≥ 𝑎𝑛;𝑛−1 ≥ ··· ≥ 𝑎𝑛;1 ≥ 𝑎𝑛;0 ≥ 0. (12)
Inparticular,if(iii)issatisfied,then𝜙𝑛̸∈𝒫𝑛;ar,althoughallzerosof 𝜙𝑛
belongtoDand0≤𝑒𝑛<𝑚isactuallyequivalentto (11).
(c) Supposethat 𝜙𝑛−1(0)=0. Then𝑒𝑛 =0 holds if andonly if the coefficients
𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛ofthepolynomial𝜙𝑛 arerealand (12)holds.Inparticular,
if 𝑒𝑛=0issatisfied,then𝑎𝑛;1=𝑎𝑛;0=0andallzerosof𝜙𝑛 belongtoD. Proof. InviewofLemma3.1weseethat(ii)implies(i).Now,wesupposethat(i) holds.By𝜙𝑛−1∈𝒫𝑛−1;arand𝜙𝑛−1(0)=𝑑𝑛−1,wehavetherepresentation
𝜙𝑛−1(𝑥) = 𝑛−1
∑︁
𝑗=0
(𝑗+1)𝑑𝑛−1𝑥𝑗.
Hence(cf.Remark2.1),itfollowsthat
𝜙𝑛(𝑥) = 𝑛 ∑︁ 𝑗=0 𝑗𝑑𝑛−1+𝑒𝑛(𝑛−𝑗)𝑑𝑛−1 √︀ 1−𝑒2𝑛 𝑥𝑗, where (𝑗+1)𝑑𝑛−1+𝑒𝑛(𝑛−𝑗−1)𝑑𝑛−1− (︀ 𝑗𝑑𝑛−1+𝑒𝑛(𝑛−𝑗)𝑑𝑛−1 )︀ = (1−𝑒𝑛)𝑑𝑛−1
for𝑗=0,1,...,𝑛−1,i.e.weget(ii).Therefore,(i)and(ii)areequivalent.In par-ticular,if(i)issatisfied,then(ii)aswellsothatthecoefficients𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛
ofthepolynomial𝜙𝑛 arerealand(11)holds.Thus,(a)isproven.
Wenowsupposethat𝜙𝑛−1(0)̸=0and𝜙𝑛−1(0)̸=𝑑𝑛−1 hold.Inparticular,
dueto(a),wehave𝜙𝑛̸∈𝒫𝑛;ar.Recallingthechoiceofthenumber𝑚according
to(b)and𝜙𝑛−1∈𝒫𝑛−1;ar,theequivalenceof(iii)and(iv)isaconsequenceof[2,
Proposition5.4(a)].Furthermore,from[2,Proposition5.4(e)]and𝜙𝑛−1(0)̸=0
we seethat 0≤𝑒𝑛 <𝑚isactually equivalentto(11).Taking 𝜙𝑛−1 ∈𝒫𝑛−1;ar
andCorollary3.1intoaccount,by[2,Proposition2.5(a)]andthechoiceof𝜙𝑛 it
followsthatallzerosof𝜙𝑛 belongtoD.Thus,(b)isproven.
Finally,wesupposethat𝜙𝑛−1(0)=0holds.Notethat𝑒𝑛=0and(1)imply
𝜙𝑛(𝑥) = 𝑥𝜙𝑛−1(𝑥).
Hence,if 𝑒𝑛 =0, thenthecoefficients 𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛 of thepolynomial 𝜙𝑛
are real and (12) holds, where 𝑎𝑛;1 = 𝑎𝑛;0 = 0 (since 𝜙𝑛−1 ∈ 𝒫𝑛−1;ar and
belong to D. Therefore, all zeros of 𝜙𝑛 belong to D when 𝑒𝑛 = 0. Using [2,
Proposition5.4(a)]onecanalsoseethat,ifthecoefficients𝑎𝑛;0,𝑎𝑛;1,...,𝑎𝑛;𝑛of
thepolynomial𝜙𝑛 arerealand(12)holds,then𝑒𝑛=0.
Asanasideto(11)and(12),wenotethat𝑎𝑛;0 =0isequivalentto𝑒𝑛 =0(cf.
Remark2.2).Furthermore,if𝜙𝑛−1(0)=0,then(c)ofProposition3.2showsthat
thereisonlythechoice𝑒𝑛=0whichrealize(12)andnochoicewhichrealize(11).
Acknowledgment: ThesecondauthorhastothankFelixKlemdforsupportduring thepreparationoftheseresults.
References
[1] N. Anderson,E.B. Saff, andR.S. Varga, An extensionof theEneström– Kakeyatheoremanditssharpness,SIAMJ.Math.Anal.12(1981),10–22. [2] A.E.ChoqueRiveroandA.Lasarow,TheEneström–Kakeyatheorem encoun-tersthetheoryoforthogonalpolynomialsontheunitcircle,Linear Algebra Appl.439(2013),1258–1285.
[3] Ph.Delsarte,Y.V.Genin,andY.G.Kamp,Orthogonalpolynomialmatrices ontheunitcircle,IEEETrans.CircuitsandSystems25(1978),149–160. [4] V.K.Dubovoj,B.Fritzsche,andB.Kirstein,MatricialVersion ofthe
Clas-sicalSchurProblem,Teubner-TextezurMath.129,Teubner,Leipzig1992. [5] A.Lasarow,Onsomespecialpolynomialsappearinginorthogonalsystems
ontheunitcircle,Analysis34(2014),353–378.
[6] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical theory,Amer. Math.Soc.Coll.Publ.54,American MathematicalSociety, Providence,2005.
[7] G.Szegő,OrthogonalPolynomials,Amer.Math.Soc.Colloq.Publ.23, Amer-icanMathematicalSociety,Providence,1939.
