Asymptotics of Laurent Polynomials of Even Degree Orthogonal with Respect to Varying Exponential Weights

1.2 MB·English

by  K. T. -R. McLaughlin

Asymptotics of Laurent Polynomials of Even Degree
Orthogonal with Respect to Varying Exponential Weights
6 K. T.-R. McLaughlin∗ A.H. Vartanian†
0
Departmentof Mathematics Department ofMathematics
0
2 TheUniversity of Arizona University ofCentral Florida

n 617N.Santa Rita Ave. P. O. Box161364
a
J P. O. Box 210089 Orlando, Florida 32816–1364

3 Tucson, Arizona85721–0089 U. S. A.
1
U.S. A.

]
A X. Zhou
‡
C
DepartmentofMathematics
.
h Duke University
t
a Box 90320
m
Durham, North Carolina 27708–0320
[
U.S. A.
1
v
6 11 January2006
0
3
1
0
Abstract
6
0 Let ΛR denote the linear space over R spanned by zk, k Z. Define the real inner product(with
∈
h/ varyingexponentialweights)h···,···iL: ΛR×ΛR→R,(f,g)7→ R f(s)g(s)exp(−NV(s))ds,N∈N,where
theexternalfieldVsatisfies:(i)VisrealanalyticonR 0;(Rii)lim (V(x)/ln(x2+1))=+ ;and(iii)
at lim (V(x)/ln(x 2+1))=+ .Orthogonalisationofth\e{(o}rdered)|bx|a→s∞e 1,z 1,z,z 2,z2,...,∞z k,zk,...
m |x|→0 − ∞ { − − − }
with respect to , L yields the even degree and odd degree orthonormal Laurent polynomials
iv: {thφem(ezv)}e∞mn=0d:φeg2nr(eze)h=···anξ···id(−2nn)ozd−nd+·d·e·+gξr(ne2ne)zmn,oξn(n2inc)>o0rt,haongdoφn2anl+1L(za)u=reξn−(2tnn−+p11o)zl−ynn−o1+m··i·a+lξsn(:2nπππ+21n)z(nz,)ξ:=−(2nn−(+ξ11)(n2>n)0)−.1Dφe2fin(nze)
X and πππ2n+1(z):=(ξ(2nn+11))−1φ2n+1(z). Asymptotics in the double-scaling limit as N,n→∞ such that
r N/n=1+o(1)ofπππ− (−z)(intheentirecomplexplane),ξ(2n),φ (z)(intheentirecomplexplane),and
a 2n n 2n
Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence
c = skexp( NV(s))ds areobtainedbyformulatingtheevendegreemonicorthogonalLaurent
k R − k Z
npolynRomial problem as ao∈matrix Riemann-Hilbert problem on R, and then extracting the large-
n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developedin[2,3].
2000MathematicsSubjectClassification.(Primary)30E20,30E25,42C05,45E05,
47B36:(Secondary)30C15,30C70,30E05,30E10,31A99,41A20,41A21,41A60
AbbreviatedTitle.AsymptoticsofEvenDegreeOrthogonalLaurentPolynomials
KeyWords.Asymptotics,equilibriummeasures,Hankeldeterminants,Laurentpolynomials,
E-mail: [email protected]
∗
E-mail: [email protected]
†
E-mail: [email protected]
‡
1
2 K.T.-R.McLaughlin,A.H.Vartanian,andX.Zhou
Laurent-Jacobimatrices,Padéapproximants,parametrices,Riemann-Hilbertproblems,sing-
ularintegralequations,strongmomentproblems,variationalproblems
AsymptoticsofEvenDegreeOrthogonalLaurentPolynomials 3
1 Introduction and Background
ConsidertheclassicalStieltjes(resp.,classicalHamburger)momentproblem(SMP)(resp.,HMP):givena
simply-infinite(moment)sequenceofrealnumbers c :
{ n}∞n=0
(i) findnecessaryandsufficientconditionsfortheexistenceofanon-negativeBorelmeasureµS
MP
+
(resp.,µH )on[0,+ )(resp.,( ,+ )),andwithinfinitesupport,suchthatc = ∞tndµS (t),
MP ∞ −∞ ∞ n 0 MP
n Z+ := 0 N (resp., c = +∞tndµH (t), n Z+), where the (improper) inRtegral is to be
∈ 0 { }∪ n MP ∈ 0
understoodintheRiemann-StiRe−l∞tjessense;
(ii) when there is a solution of the existence problem, in which case the SMP (resp., HMP) is
determinate,findconditionsfortheuniquenessofthesolution;and
(iii) when there is more than one solution, in which case the SMP (resp., HMP) is indeterminate,
describethefamilyofallsolutions.
The SMP was—first—treated in 1894/95 by Stieltjes in the pioneering works [4], and the HMP
was introduced and solved in 1920/21 by Hamburger in the landmark works [5]. The subsequent
developmentofthetheoryofmomentproblemsbroughtforththeprofoundfactthat,overandabove
theindispensableutilityaffordedbytheanalytictheoryofcontinuedfractions,inparticular,S-and
realJ-fractions,thetheoryoforthogonalpolynomials[6]playedaseminal,intimateandcentralrôle
(see,forexample,[7]).
Questionsregardingtwosimply-infinite(moment)sequences cn n Z+ and c n n N ofrealnum-
{ } ∈ 0 { − } ∈
bers, or, equivalently, doubly- or bi-infinite (moment) sequences cn n Z of real numbers, manifest,
in various settings, purely mathematical and/or otherwise, as na{tur}al∈extensions of the foregoing.
This generalisation is colloquially refered to as the strong Stieltjes (resp., strong Hamburger) moment
problem (SSMP)(resp.,SHMP),namely, given a doubly- or bi-infinite (moment) sequence cn n Z of
{ } ∈
realnumbers:
(1) findnecessaryandsufficientconditionsfortheexistenceofanon-negativemeasureµSS (resp.,
MP
+
µSH)on[0,+ )(resp.,( ,+ )),andwithinfinitesupport,suchthatc = ∞tndµSS(t),n Z
MP ∞ −∞ ∞ n 0 MP ∈
(resp.,c = +∞tndµSH(t),n Z),wherethe(improper)integralistobeundRerstoodinthesense
n MP ∈
ofRiemannR−-S∞tieltjes;
(2) whenthereisasolution,inwhichcasetheSSMP(resp.,SHMP)isdeterminate,findconditions
fortheuniquenessofthesolution;and
(3) whenthereismorethanonesolution,inwhichcasetheSSMP(resp.,SHMP)isindeterminate,
describethefamilyofallsolutions.
TheSSMP(resp.,SHMP)wasintroducedin1980(resp.,1981)byJonesetal.[8](resp.,Jonesetal.[9]),
andstudiedfurtherin[10–14](see,also,thereviewarticle[15]).UnlikethemomenttheoryfortheSMP
and the HMP, wherein the theory of orthogonal polynomials, and the analytic theory of continued
fractions,enjoyedaprominentrôle,theextensionofthemomenttheorytotheSSMPandtheSHMP
introduceda‘rationalgeneralisation’of theorthogonal polynomials, namely,the orthogonalLaurent
(or L-) polynomials (as well as the introduction of special kinds of continued fractions commonly
referredtoaspositive-Tfractions),whicharediscussedbelow[10–21].(TheSHMPcanalsobesolved
usingthespectraltheoryofunboundedself-adjointoperatorsinHilbertspace[22];see,also,[23].)
For any pair (p,q) Z Z, with p6q, let ΛC := f: C C; f(z)= q λ zk, λ C, k=p,...,q ,
∈ × p,q ∗→ k=p k k∈
(cid:26) (cid:27)
AwhfuernectCio∗n:=(Cor\e{l0e}m.Feonrt)anfyΛmC∈iZsc+0a,lsleetdΛaC2mLa:=urΛenC−tm(,mo,rΛL-C2)m+p1ol:=ynΛomC−mi−al1.,m(N,aonPtde:ΛthCbe:=se∪tsmb∈ΛZC+0(ΛanC2md∪ΛΛCC2fmo+r1m).
∈ p,q
linearspacesoverthefieldCwithrespecttotheoperationsofadditionandmultiplicationbyascalar.)
Bases for each of the spaces ΛC , ΛC , and ΛC, respectively, are z m,...,zm , z m 1,...,zm , and
2m 2m+1 { − } { − − }
const.,z 1,z,z 2,z2,...,z k,zk,... (thebasisforΛCcorrespondstothecyclically-repeatedpolesequence
− − −
{nopole,0, ,0, ,...,0, ,... ).}Furthermore, note that, for each 0. f ΛC, there exists a unique
l{ Z+ sucht∞hat f∞ΛC.Fo∞rl Z}+ and0.f ΛC,theL-degreeof f,symbolic∈allyLD(f),isdefinedas
∈ 0 ∈ l ∈ 0 ∈ l
LD(f):=l.
4 K.T.-R.McLaughlin,A.H.Vartanian,andX.Zhou
ForΛC f= λ zj,setC (f):=λ , j Z.Foreachl Z+ and0.f ΛC,definetheleadingcoefficient
of f,sym∋bolicaj∈lZlyLjC(f),andj thetrajilin∈gcoefficientof∈f,sy0mbolically∈TCl(f),asfollows:
P
b b
λ , l=2m,
LC(f):= m
λ , l=2m+1,
b−m−1
and b
λ , l=2m,
TC(f):= −m
λ , l=2m+1.
bm
Thus,forl∈Z+0 and0.f∈ΛCl ,onewrites,for fb:=fl(z):(1)ifl=2m,
f (z)=TC(f)z m+ +LC(f)zm;
2m −
···
and(2)ifl=2m+1,
f2m+1(z)=LC(f)z−m−1+ +TC(f)zm.
···
Forl Z+,0.f ΛCiscalledmonicifLC(f)=1.
∈ 0 ∈ l
Consider the positive measure on R (oriented throughout this work, unless stated otherwise,
from to+ )givenby
−∞ ∞
dµ(z)=w(z)dz,
withvaryingexponentialweightfunctionoftheform
e e
w(z)=exp( NV(z)), N N,
− ∈
wheretheexternalfieldV: R 0 Rsatisfiesthefollowingconditions:
e
\{ }→
V isrealanalyticon R 0 ; (V1)
\{ }
lim V(x)/ln(x2+1) =+ ; (V2)
x ∞
| |→∞(cid:16) (cid:17)
lim V(x)/ln(x−2+1) =+ . (V3)
x 0 ∞
| |→ (cid:16) (cid:17)
(For example, a rational function of the form V(z) = 2m2 ̺ zk, with ̺ R, k = 2m ,...,2m ,
m1,2 ∈ N, and ̺−2m1,̺2m2 > 0 would suffice.) Define (uPnki=q−u2eml1y)kthe strongkm∈oment li−near1functiona2l
L by its action on the basis elements of ΛC: L: ΛC ΛC, f = λ zk L(f) := λ c ,
where ck = L(zk) = Rskexp(−NV(s))ds, (k,N) ∈ Z × N→. (Note thaPt,ka∈sZbpker t7→he discussioPnk∈aZbbokvek,
c = skexp( NV(sR))ds, N N is a bi-infinite, real-valued, strong moment sequence: c is called
k R − ∈ k Z k
nthekRthstrongmomentofL.)Assocoia∈tedwiththeabove-definedbi-infinite,real-valued,strongmoment
sequence{ck}k∈ZaretheHankeldeterminantsHk(m),(m,k)∈Z×N[10,11,15,17]:
cm cm+1 cm+k 2 cm+k 1
··· − −
(cid:12) cm+1 cm+2 cm+k 1 cm+k (cid:12)
(cid:12) ··· − (cid:12)
H0(m):=1 and Hk(m):=(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cm...+2 cm...+3 ·.·.·. cm...+k cm+...k+1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12). (1.1)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)cm+k−1 cm+k ··· cm+2k−3 cm+2k−2(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
Foranypair(p,q) Z Z,withp6q,letΛR :=(cid:12) f: C C; f(z)= q λ zk, λ R,(cid:12)k=p,...,q ,and
∈ × p,q (cid:26) ∗→ k=p k k∈ (cid:27)
(dNefiontee:,tahnealsoegtsouΛsRlyaasnadboΛvRe,ffoorrmml∈inZe+0a,rΛsR2pma:c=eΛsR−omv,mer,ΛthR2me+fi1:e=ldΛRR−m−wP1,imt,haenredspΛeRect:=to∪mt∈hZe+0(oΛpR2emra∪tiΛoR2nms+1o)f.
p,q
additionandmultiplicationbyascalar;furthermore,ΛR ( ΛC)isthelinearspaceoverRspanned
byzj, j Z.)Hereafter,weshallbeconcernedonlywith(re⊂al)L-polynomialsinΛR:the—ordered—
basefor∈ΛR is 1,z 1,z,z 2,z2,...,z k,zk,... ,correspondingtothecyclically-repeatedpolesequence
− − −
nopole,0, ,0{, ,...,0, ,... .Definethe}realbilinearform , Lasfollows: , L: ΛR ΛR R,
{(f,g)7→hf,g∞iL:=∞L(f(z)g(∞z))=}R f(s)g(s)e−NV(s)ds, N∈N. It ihs···a···ifact[10,11,15h,···17···i] that th×e bili→near
form , L thus defined is anRinner product if and only if H(−2m)>0 and H(−2m)>0 m Z+ (see
h··· ···i 2m 2m+1 ∀ ∈ 0
Equations(1.8)below,andSubsection2.2,theproofofLemma2.2.1);andthisfactisused,withlittle
ornofurtherreference,throughoutthiswork(see,also,[24]).
AsymptoticsofEvenDegreeOrthogonalLaurentPolynomials 5
Remark1.1. Theselattertwo(Hankeldeterminant)inequalitiesalsoappearwhenthequestionofthe
solvabilityof the SHMPisposed(inthis case,the c , k Z,which appearinEquations (1.1)should
k
be replacedby cSHMP, k Z): indeed, if these two inequa∈lities are true m Z+, then there is a non-
k ∈ ∀ ∈ 0
negativemeasureµSH (onR)withthegiven(real)moments.ForthecaseoftheSSMP,therearefour
MP
(Hankel determinant) inequalities (in this latter case, the c , k Z, which appear in Equations (1.1)
k
∈
shouldbereplacedbycSSMP,k Z)whichguaranteetheexistenceofanon-negativemeasureµSS (on
k ∈ MP
[0,+ ))withthegivenmoments,namely[8](see,also,[10,11]):foreachm Z+,H(−2m)>0,H(−2m)>0,
∞ ∈ 0 2m 2m+1
H(−2m+1)>0, and H(−2m−1)<0. It is interesting to note that the former solvability conditions do not
2m 2m+1
asiumtoilmarastitcaatellmyeimntphlyoltdhsattrtuheefpoorstihtievSeM(rePa(ls)emeothmeelnattste{rcSkfHoMuP}rk∈sZo+0lvdaebtielirtmyicnoenadimtieoanssu).reviatheHMP:(cid:4)a
If f ΛR,then
∈ f() L:=( f, f L)1/2
k · k h i
is called the norm of f with respect to L: note that f() L>0 f ΛR, and f() L>0 if 0. f ΛR.
k · k ∀ ∈ k · k ∈
t{oφ♭nL(zi)f},n∈Zm+0 i,sncaZlle+d: a(real)orthonormalLaurent(orL-)polynomialsequence(ONLPS)withrespect
∀ ∈ 0
(i) φ♭ ΛR,thatis,LD(φ♭):=n;
n∈ n n
(ii) hφ♭m,φ♭n′iL=0 ∀ m,n′,or,alternatively,hf,φ♭niL=0 ∀ f∈ΛRn−1;
(iii) hφ♭m,φ♭miL=:kφ♭m(·)k2L=1.
Orthonormalisation of 1,z 1,z,z 2,z2,...,z n,zn,... , correspondingto the cyclically-repeatedpole
− − −
{ }
sequence nopole,0, ,0, ,...,0, ,... , with respect to , L via the Gram-Schmidt orthogonal-
{ ∞ ∞ ∞ } h··· ···i
isation method, leads to the ONLPS, or, simply, orthonormal Laurent (or L-) polynomials (OLPs),
φm(z) m Z+,which,bysuitablenormalisation,maybewrittenas,form=2n,
{ } ∈ 0
φ2n(z)=ξ(−2nn)z−n+···+ξn(2n)zn, ξn(2n)>0, (1.2)
and,form=2n+1,
φ2n+1(z)=ξ(2nn+11)z−n−1+···+ξn(2n+1)zn, ξ(2nn+11)>0. (1.3)
− − − −
Theφ ’sarenormalisedsothattheyallhaverealcoefficients;inparticular,theleadingcoefficients,
n
LC(φ2n):=ξ(n2n) andLC(φ2n+1):=ξ(2nn+11), n∈Z+0,arebothpositive, ξ(00)=1,andφ0(z)≡1.Eventhough
the leading coefficients, ξ(2n) and−ξ−(2n+1), n Z+, are non-zero (in particular, they are positive), no
n n 1 ∈ 0
sFuucrhthreersmtroicreti,onnotaeptphlaiets,btyoctohnesttrrauiclit−niog−nc:oefficients, TC(φ2n):=ξ−(2nn) and TC(φ2n+1):=ξn(2n+1), n∈Z+0.
(1) φ2n,zj L=0, j= n,...,n 1;
h i − −
(2) φ2n+1,zj L=0, j= n,...,n;
h i −
(3) hφj,φkiL=δjk, j,k∈Z+0,whereδjk istheKroneckerdelta.
Moreover,if, for eachm∈Z+0, the orthonormal L-polynomials φ2m(z) and φ2m+1(z), respectively, are
suchthatTC(φ2m):=ξ(2mm),0andTC(φ2m+1):=ξ(m2m+1),0,thentherearespecialChristoffel-Darboux
formulaefortheOLPs−(see,forexample,[12,17];see,also,[25]):
φ2m(ζ)(zφ2m−1(z)−ζφ2m−1(ζ))−ζφ2m−1(ζ)(φ2m(z)−φ2m(ζ))=(z−ζ)ξξ(2−(mm2mm−)1) 2Xmj=−01φj(z)φj(ζ),
−
ξ(2m+1) 2m
φ2m(ζ)(zφ2m+1(z)−ζφ2m+1(ζ))−ζφ2m+1(ζ)(φ2m(z)−φ2m(ζ))=(z−ζ) ξmm(2m) Xj=0 φj(z)φj(ζ),
whereφ (z) 0,and(dividingbyz ζandlettingζ z)
1
− ≡ − →
φ2m(z)ddz(zφ2m−1(z))−zφ2m−1(z)ddzφ2m(z)=ξξ(2−(mm2mm−)1) 2Xmj=−01(φj(z))2,
−
6 K.T.-R.McLaughlin,A.H.Vartanian,andX.Zhou
d d ξ(2m+1) 2m
φ2m(z)dz(zφ2m+1(z))−zφ2m+1(z)dzφ2m(z)= ξmm(2m) Xj=0(φj(z))2.
Itisconvenienttointroducethemonic orthogonalLaurent(orL-)polynomials, πππ (z), j Z+:(i)
for j=2n,n Z+,withπππ (z) 1, j ∈ 0
∈ 0 0 ≡
πππ2n(z):=φ2n(z)(ξ(n2n))−1=ν−(2nn)z−n+···+zn, ν−(2nn):=ξ−(2nn)/ξn(2n); (1.4)
and(ii)for j=2n+1,n Z+,
∈ 0
πππ2n+1(z):=φ2n+1(z)(ξ(2nn+11))−1=z−n−1+···+νn(2n+1)zn, νn(2n+1):=ξn(2n+1)/ξ(2nn+11). (1.5)
− − − −
ThemonicorthogonalL-polynomials,πππ (z), j Z+,possessthefollowingproperties:
j ∈ 0
(1) πππ2n,zj L=0, j= n,...,n 1;
h i − −
(2) πππ2n+1,zj L=0, j= n,...,n;
h i −
(3) hπππ2n,πππ2niL=:kπππ2n(·)k2L=(ξ(n2n))−2,whenceξn(2n)=1/kπππ2n(·)kL(>0);
(4) hπππ2n+1,πππ2n+1iL=:kπππ2n+1(·)k2L=(ξ(2nn+11))−2,whenceξ(2nn+11)=1/kπππ2n+1(·)kL(>0).
− − − −
Furthermore, in terms of the Hankel determinants, H(m), (m,k) Z N, associated with the real-
k ∈ ×
valued, bi-infinite, strong moment sequence c = ske NV(s)ds, N N , the monic orthogonal
k R − ∈ k Z
L-polynomials,πππ (z), j Z+,arerepresentedviantheRfollowingdeterminaont∈alformulae[10,11,15,17]
(see,also,Subsectjion2.∈2,P0roposition2.2.1):form Z+,
∈ 0
c 2m c 2m+1 c 1 z−m
(cid:12)c −2m+1 c−2m+2 ··· c−0 z−m+1(cid:12)
πππ2m(z)=H2(−1m2m) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −c... −c... ·.·.·. c ... zm... 1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12), (1.6)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c−01 c10 ······ c22mm−−12 zm− (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and
c c c z m 1
2m 1 2m 1 − −
(cid:12) −c 2m− c −2m+1 ··· c−0 z−m (cid:12)
πππ2m+1(z)=−H2(−1m2+m1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c−... −c... ·.·.·. c ... zm... 1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12); (1.7)
(cid:12)(cid:12)(cid:12)(cid:12) c−01 c01 ······ 2cm2m−1 zm− (cid:12)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
moreover,itcanbeshownthat(see,forex(cid:12)ample,[15,17]),forn Z+, (cid:12)
∈ 0
ξ(n2n) =kπππ2n1(·)kL!=vutHH22((−−nn22+nn1)), ξ−(2nn−+11) =kπππ2n+11(·)kL!=vutHH2(−n2(−2+nn2+2−n1)2), (1.8)
Foreachm Z+,νth(−2nen)m:=onξξi(n(−c22nnno))rt=hoHgH2(o−n2(n−2nn2a+nl)1L),-polynoνmn(2ina+l1)πππ:=(zξξ)−(n(a22nnnn−++d111))th=e−inHdH2(e−n2(x−2+nn2+1m−n1)1a).recallednon-sing(u1l.a9r)
∈ 0 m
ν(2n), m=2n,
if0,TC(πππm):=ν(−2nn+1), m=2n+1; otherwise,πππm(z)andmaresingular.FromEquations(1.9),itcan
beseenthat,foreanchm∈Z+0:
(i) πππ2m(z)isnon-singular(resp.,singular)ifH2(−m2m+1),0(resp.,H2(−m2m+1)=0);
AsymptoticsofEvenDegreeOrthogonalLaurentPolynomials 7
(ii) πππ2m+1(z)isnon-singular(resp.,singular)ifH2(−m2+m1−1),0(resp.,H2(−m2+m1−1)=0).
F[1o0r,e1a1c,1h7m]t∈hZat+0,,folertmµ2mZ:=+:card{z; πππ2m(z)=0}andµ2m+1:=card{z; πππ2m+1(z)=0}.Itisanestablishedfact
∈ 0
(1) the zeros of πππ (z) are real, simple, and non-zero, and µ = 2m (resp., 2m 1) if πππ (z) is
2m 2m 2m
−
non-singular(resp.,singular);
(2) thezerosofπππ2m+1(z)arereal,simple,andnon-zero,andµ2m+1=2m+1(resp.,2m)ifπππ2m+1(z)is
non-singular(resp.,singular).
For each m Z+, it can be shown that, via a straightforward factorisation argument and using
∈ 0
Equations(1.6)and(1.7):
(i) ifπππ (z)isnon-singular,uponsetting α(2m), k=1,...,2m := z; πππ (z)=0 ,
2m k { 2m }
n o
2m
α(2m)=ν(2m);
k m
Yk=1 −
(ii) ifπππ2m+1(z)isnon-singular,uponsetting α(k2m+1), k=1,...,2m+1 :={z; πππ2m+1(z)=0},
n o
2m+1
α(2m+1)= ν(2m+1) −1.
k − m
Yk=1 (cid:16) (cid:17)
Unlikeorthogonalpolynomials,whichsatisfyasystemofthree-termrecurrencerelations,monic
orthogonal,andorthonormal,L-polynomialsmaysatisfyrecurrencerelationsconsistingofapairof
four-termrecurrencerelations[15],apairofsystemsofthree-orfive-termrecurrencerelations(which
isguaranteedinthecasewhenthecorrespondingmonicorthogonal,andorthonormal,L-polynomials
arenon-singular)[15–17],orasystemconsistingoffourfive-termrecurrencerelations[23].
Remark1.2. Thenon-vanishing of theleadingandtrailingcoefficientsofthe OLPs φ (z) ,that
{ m }∞m=0
is,
ξ(2n), m=2n,
LC(φ ):= n
m ξ(2n+1), m=2n+1,
 −n−1
and 
ξ(2n), m=2n,
TC(φm):=ξ(−2nn+1), m=2n+1,
 n
respectively,isofparamountimportance:ifboththeseconditionsarenotsatisfied,thenthe‘length’
oftherecurrencerelationsmaybegreaterthanthree[16](see,also,[24]). (cid:4)
It can be shown that (see, for example, [17], and Chapter 11 of [26]), if πππm(z) m Z+, as defined
above,isanon-singular, monic orthogonal L-polynomialsequence,thatis, H{(−2n+1)},∈00(m=2n)and
2n
H2(−n2+n1−1),0(m=2n+1),then{πππm(z)}m∈Z+0 satisfythepairofthree-termrecurrencerelations
z 1
πππ2m+1(z)=β−♮ +β♮2m+1πππ2m(z)+λ♮2m+1πππ2m−1(z),
 2mz 
πππ2m+2(z)=β♮ +β♮2m+2πππ2m+1(z)+λ♮2m+2πππ2m(z),
 2m+1 
whereπππ (z) 0,
1
− ≡
β♮ =ν(2m), β♮ =ν(2m+1),
2m m 2m+1 m
−
λ♮2m+1=−HH2(−m(−2+m21m−)1H)H(−2(−2mm2−m+1+1)2) (,0), λ♮2m+2=−HH2((−−m22+mm2)−H1)(H−22(m−m2−m1)) (,0),
2m 2m 2m+1 2m+1
8 K.T.-R.McLaughlin,A.H.Vartanian,andX.Zhou
andλ β /β >0 j N,withλ := c ,leadingtoatri-diagonal-typeLaurent-Jacobimatrix forthe
j j 1 j 1 1
− ∀ ∈ − − F
‘mixed’mapping
F: ΛR→ΛR, f(z)7→(z−1(⊕∞n=0diag(1,0))+z(⊕∞n=0diag(0,1)))f(z),
where diag(1,0):=diag(1,0,...,1,0,...),and diag(0,1):=diag(0,1,...,0,1,...),
⊕∞n=0 ⊕∞n=0
β♮ 1
− 1
λ♮ β♮ 1
− 2 − 2 
F = diag(cid:16)β♮0,β♮1,β♮2,...(cid:17) −λ♮3 −−λβ♮3♮4 −−1λβ♮4♮5 −−1λβ♮5♮6 −.1.β.♮6 −λ.♮21.m.+1 −−λβ.♮2♮2.mm.++12 −β.♮21.m.+2 .1.. ... ,
withzerosoutsidetheindicateddiagonals(intermsof φm(z) m Z+,thepairofthree-termrecurrence
{ } ∈ 0
relationsreads[16]:
φ2m+1(z)=(z−1+g2m+1)φ2m(z)+f2m+1φ2m 1(z),
−
φ2m+2(z)=(1+g2m+2z)φ2m+1(z)+f2m+2φ2m(z),
pwahireroefffi2mv+e1-,tef2rmm+2r,ec0u,rmren∈cZe+0re,lφa−ti1o(nz)s≡[107,]a,nwdithφ0πππ(z)(≡z)1);0o,tjh=e1rw,2i,se, {πππm(z)}m∈Z+0 satisfythe following
j
− ≡
πππ2m+2(z)=γ♭2m+2,2m−2πππ2m−2(z)+γ♭2m+2,2m−1πππ2m−1(z)+(z+γ♭2m+2,2m)πππ2m(z)
+γ♭2m+2,2m+1πππ2m+1(z),
πππ2m+3(z)=γ♭2m+3,2m 1πππ2m 1(z)+γ♭2m+3,2mπππ2m(z)+(z−1+γ♭2m+3,2m+1)πππ2m+1(z)
− −
+γ♭2m+3,2m+2πππ2m+2(z),
where γ = 0, k < 0, l > 2, leading to a penta-diagonal-type Laurent-Jacobi matrix for the ‘mixed’
l,k
G
mapping
G: ΛR→ΛR, g(z)7→(z(⊕∞n=0diag(1,0))+z−1(⊕∞n=0diag(0,1)))g(z),
γ♭ γ♭ 1
− 2,0 − 2,1
γ♭ γ♭ γ♭ 1
− 3,0 − 3,1 − 3,2 
G=−γ♭4,0 −−γγ♭4♭5,,11 −−−γγγ♭4♭5♭6,,,222 −−−.γγγ.♭4♭5♭6.,,,333 −γ♭2−−m.γγ+1.2♭5♭6.,,,244m−2 −−γγ♭2♭2−mm.γ++1.♭632.,,,522mm−−11 −−γγ♭2♭2..mm1..++..32,,22mm −−γγ♭2♭2mm..++..32..,,22mm++11 −γ♭2m.+1.3.,2m+2 .1.. ... ,
withzerosoutsidetheindicateddiagonals.Thegeneralformofthese(systemof)recurrencerelations
isapairofthree-andfive-termrecurrencerelations[23]:forn Z+,
∈ 0
zφ2n+1(z)=b♯2n+1φ2n(z)+a♯2n+1φ2n+1(z)+b♯2n+2φ2n+2(z),
zφ2n(z)=c♯2nφ2n−2(z)+b♯2nφ2n−1(z)+a♯2nφ2n(z)+b♯2n+1φ2n+1(z)+c♯2n+2φ2n+2(z),
whereallthecoefficientsarereal,c♯=b♯=0,andc♯ >0,k N,and
0 0 2k ∈
z−1φ2n(z)=β♯2nφ2n−1(z)+α♯2nφ2n(z)+β♯2n+1φ2n+1(z),
AsymptoticsofEvenDegreeOrthogonalLaurentPolynomials 9
z−1φ2n+1(z)=γ♯2n+1φ2n−1(z)+β♯2n+1φ2n(z)+α♯2n+1φ2n+1(z)+β♯2n+2φ2n+2(z)+γ♯2n+3φ2n+3(z),
whereallthecoefficients arereal,β♯=γ♯=0,β♯>0,andγ♯ >0,l N,leading, respectively,tothe
0 1 1 2l+1 ∈
real-symmetric,tri-penta-diagonal-typeLaurent-Jacobimatrices, and ,forthemappings
J K
J: ΛR ΛR, j(z) zj(z) and K: ΛR ΛR, k(z) z−1k(z),
→ 7→ → 7→
a♯ b♯ c♯
0 1 2
b♯ a♯ b♯ 
J =c♯21 b1♯2 bac♯42♯2♯3 bba♯3♯4♯3 bbacc♯6♯4♯4♯5♯4 bba♯5♯5♯6 babcc♯6♯8♯6♯6♯7 bba♯7♯7♯8 bc♯2♯2.backk.♯8♯8♯8++.21 ab♯2♯2.bkk.♯9++.12 bac♯2♯2..kk♯1..++0..22 b♯2.k.+.3 c♯2.k.+.4 ,
and
α♯ β♯
0 1
β♯ α♯ β♯ γ♯ 
K = 1 γβ♯21♯3 αβ2♯3♯2 αγββ♯3♯43♯3♯5 αββ♯4♯5♯4 γαγββ♯5♯6♯5♯7♯5 α.βββ.♯8♯7♯6♯6. ααγ.β.♯7♯7♯8♯7. γ♯2.ββk.♯9♯8+.1 β♯2γk♯9+1 αβ♯2.♯2kk.++.21 αβ♯2.♯2kk.++.22 γβ♯2.♯2kk.++.33 ,
withzerosoutsidetheindicateddiagonals;moreover,asshownin[23], and areformalinverses,
J K
thatis, = =diag(1,...,1,...)(see,also,[27–31]).
JK KJ
Itisconvenientatthispointtodiscuss,ifonlysuccinctly,afewofthemultitudinousapplications
ofL-polynomials(completedetailsmaybefoundintheindicatedreferences):
(1) as stated at the beginning of the Introduction, L-polynomials are intimately related with the
solution of the SSMP and the SHMP. It is important to note [14] that the classical and strong
moment problems (SMP, HMP, SSMP, and SHMP) are specialcases of a more general theory,
wheremomentscorrespondingtoanarbitrary,countablesequenceof(fixed)pointsareinvolved
(in the classical and strong moment cases, respectively, the points are repeated and 0,
∞ ∞
cyclicallyrepeated),andwhereorthogonalrationalfunctions[26,32,33]playtherôleoforthogonal
polynomialsandorthogonalLaurent(orL-)polynomials;furthermore,sinceL-polynomialsare
rationalfunctionswith(fixed)polesattheoriginandatthepointatinfinity, thesteptowards
amoregeneraltheorywherepolesareatarbitrary,butfixed,positions/locations inC is
∪{∞}
natural,withapplicationsto,say,multi-pointPadé,andPadé-type,approximants[24,34–38];
10 K.T.-R.McLaughlin,A.H.Vartanian,andX.Zhou
b
(2) innumericalanalysis,thecomputationofintegralsoftheform f(s)dµ(s),whereµisapositive
a
measure on [a,b], and 6a<b6+ , is an important probRlem. The most familiar quadra-
−∞ ∞
tureformulaearetheso-calledGauss-Christoffelformulae,thatis,approximatingtheintegral
b f(s)dµ(s) via a weighted-sum-of-products of function values of the form n A f(x ),
a j=1 j,n j,n
nR ∈ N, where one chooses for the nodes {xj,n}nj=1 the zeros/roots of ϕn(z), thePpolynomial of
b
degreenorthogonalwithrespecttotheinnerproduct f,g = f(s)g(s)dµ(s),andforthe(posi-
h i a
tive)weights{Aj,n}nj=1theso-calledChristoffelnumbers[35].WRhenconsideringthecomputation
ofintegralsoftheform π g(eiθ)dµ(θ),wheregisacomplex-valuedfunctionontheunitcircle
π
D:= z C; z=1 andµiRs−,say,apositivemeasureon[ π,π],inparticular,whengiscontinuous
{ ∈ | | } −
onD,keepinginmindthatafunctioncontinuousonDcanbeuniformlyapproximatedbyL-
polynomials,itisnaturaltoconsider,insteadoforthogonalpolynomials,Laurentpolynomials,
whicharealsorelatedtotheassociatedtrigonometricmomentproblem[35,39](see,also,[40]);
(3) for V: R 0 R as described by conditions (V1)–(V3), consider the function g(z)= (1+
\{ }→ R
sz) 1dµ(s), where dµ(s) =exp( NV(s))ds, N N, which is holomorphic for z C R,Rwith
−
− ∈ ∈ \
associatedasymptoticexpansions
e e
∞ ∞
g(z) = ( 1)mcmzm=:L0(z) and g(z) = ( 1)mc mz−m=:L∞(z),
C\R∋z→0 Xm=0 − C\R∋z→∞−Xm=1 − −
wherec = sle NV(s)ds,l Z,withrespecttothe(unbounded)domain z C; ε6 Arg(z) 6π ε ,
l R − ∈ { ∈ | | − }
whereArgR( )denotestheprincipalargumentof ,andε>0issufficientlysmall.Giventhepair
∗ ∗
offormalpowerseries(L0(z),L (z)),therationalfunctionP (z)/Q (z),whereP (z)belongs
∞ k,n k,n k,n
to the space of all polynomials of degree at most n 1, and Q (z) is a polynomial of degree
k,n
−
exactly n with Q (0),0, is said to be a [k/n](z) two-point Padé approximant to (L0(z),L (z)),
k,n ∞
k 0,1,...,2n ,ifthefollowingconditionsaresatisfied:
∈{ }
L0(z) Pk,n(z)(Qk,n(z))−1 = (zk),
− z 0O
→
L∞(z) Pk,n(z)(Qk,n(z))−1 = (z−1)2n−k+1 .
− z O
→∞ (cid:16) (cid:17)
The ‘balanced’ situation corresponds to the case when k = n, in which case, the two-point
Padéapproximantsaredenoted,simply,as[n/n](z).Animportant,relatedproblemofcomplex
approximationtheoryistostudytheconvergenceofsequencesoftwo-pointPadéapproximants
constructedfromthe—formal—pair(ofpowerseries)(L0(z),L (z))tothefunctiong(z)onC R;
∞
\
in particular, denoting by E (z) the ‘error term’ for the [n/n](z) approximant, that is, E (z):=
n n
g(z) [n/n](z),itcanbeshownthat,following[41],
−
E (z)= φ ( 1/z) −1 φn(s)e−NV(s) ds, z C R, (TPA1)
n (cid:16) n − (cid:17) ZR 1+sz ∈ \
where φm(z) m Z+ aretheorthonormalL-polynomialsdefinedinEquations(1.2)and(1.3).The
{ } ∈ 0
main question regarding the convergence of two-point Padé approximants for this class of
functions is with which rate it takes place, that is, the so-called quantitative result [42]: this
necessitates obtaining results for the asymptotic behaviour (as n ) of the orthonormal L-
→∞
polynomials φ (z) in the entire complex plane. The theory of orthogonal L-polynomials is a
n
natural framework for developing the theory of two-point Padé approximants, for both the
scalarandmatrixcases[24,41–44];
(4) itturnsoutthat,unlikethe(finite)non-relativisticTodalattice,whosedirectandinversespectral
transform was constructed by Moser [45], and which is based on the theory of orthogonal
polynomials and tri-diagonal Jacobi matrices, the direct and inverse scattering transform for
the (finite) relativistic Toda lattice, introduced by Ruijsenaars [46], is based on the theory of
orthogonalL-polynomialsandpairsofbi-diagonalmatrices[47](see,also,[48]);and
(5) for afinite,countable or uncountableindex setK, let ςp, p K C+:= z C; Im(z)>0 ,with
{ ∈ } ⊂ { ∈ }
ς ,ς p,q K,and ̟ , p K Cbegivenpointsets.AfunctionF(z)whichisanalyticfor
p q p
∀ ∈ { ∈ } ⊂