Some results on biorthogonal polynomials

SOME RESULTS

ON BIORTHOGONAL

POLYNOMIALS

RICHARDW.RUEDEMANN 4422W.

Greenway

Road Glendale,Arizona 85306

(Received

July8, 1993 and in revisedform October10,

1993)

ABSTRACT.

Some biorthogonal polynomials ofHahn and

Pastro

are derivedusingapolynomial modificationofthe

Lebesgue

measure dO combined withanalyticcontinuation.

A

result isgiven forchangingthe measures ofbiorthogonal polynomialsontheunit circlebythemultiplicationof theirmeasures

by

certain

Laurent

polynomials.

KEY

WORDS

AND

PHRASES: Biorthogonal polynomials,aformulaof Christoffel,changeof weight,unit circle, determinant.

1991 AMS

SUBJECT

CLASSIFICATION CODES:

Primary: 42C05 Secondary: 33A65

le

INTRODUCTION

In

[9],

Pastro

introduced apair of polynomialsetswhich arebiorthogonalontheunit circle with respecttotheweightfunction

f,z(z ;q

2)

(q2;q)(R) (abq2;q)(R)

(qz

;qZ)(R) (qz-1;q)(R)

(aq2;q)(R)(bq;q)(R)(qaz;q)(R)(qbz_;q2)(R)

z e

i

where

(t;q),,

H(1-a),

(t;q).

H(1-a).

-0 -0

To

beprecise, heshowed that if

.(z)}

and

q.(z)}

aredefined

by

p.(z) p.(z,a,b

(;q)

(b;q)"

-0

(q;q) (q;q).

_

(q-z)

and

q.(z)-q.(z,a,b)-p.(z,E

then

p.(z)q.(z)fl(z;qO

(;q).

(q;q),

q-6,,,

z-e

(1.1)

Pastro

sumetheparametersaandbarereal but as -Salam and Ismailnote

1]

they

donothave

tobe.

ese

polynomials generalize those of

key

(a

b,both

real),

and Szegfi

(a

b

0),

see

[10].

A

weit

equivalentto

z;q

)

was considered earlierbyHahn

[4]

in the caseof real coef-ficients.

Throughout

thispaperweassumethatqis realand,forconvergence ofthe infinite

products,

<

-

nsideringthe denominatorof

(z;q

)

wealsowant

[q[

<1

and[qbz-[

1, that is

<l l <l l

626 R.W. RUEDEMANN

In

section2we state, indeterminantform,apairof polynomialsetswhicharebiorthogonalon

theunit circle withrespecttothemeasure

dv(O).z-"(z_at)(z-oa)...(z-cq)dO

z.e

’

assuming that no

a

is zeroandthat 0 m h.

In

Section3weshowhow theseyield Pastro’s polynomialsin thespecialcasea

q2,,

b

Thefullresultfollows by analyticcontinuation.

Pastroalsogavein

[9]

explicit examples of Laurent orthogonal polynomials, makingconcrete

the earlier work ofJonesand Thron

[7]

in which such"polynomials"wereintroduced.

(They

are

notactually polynomials, theycontainboth positiveandnegativepowers oftheir

variable.)

More thanthis, he states an interestingconnection betweenbiorthogonal polynomials and orthogonal Laurent polynomials.

Thereis awell-known formula ofChristoffelfor modifying the measure

da(x)

by polynomial multiplication. That is, let

o(x

(x

x)

(x

x2).

.(x

x,)

beapolynomialwhichisnon-negativeon

[a,b

andlet

{q,(x)}

be thepolynomials orthogonalwith

respecttothenew measurep(x)dcx(x)on

[a,b].

Then thepolynomials

{q,,(x)}

canberepresented intermsofthepolynomials

{p,(x)}

by

p(x

)q,(x

c,

det

forsuitableconstantsc,,.

p.(x,) p./(x,)

BoththisformulaofChristoffelandarelated formulaofUvarov carryovertopolynomials

orthogonal

onthe unit circle. See Godoy and Marcellan

[3]

orIsmail and Ruedemann

[6].

The natural question is,does thisformula ofChristoffel have ananalogueforbiorthogonal polynomials onthe unit circle?

In

Section 4 we showhowatrivial modificationoftheresult in

[6]

yieldsaresult for biorthogonal polynomials, atleast for certaincases. Unfortunately, we only allow certain modificationsandmustassumethat certain determinants donotvanish. Actually,thisassumption of nonzerodeterminants iscommontobiorthogonality

(see

thework ofBaxter

[2]).

In

theremainderofthispaperwe

adopt

the followingnotation.

For p,(z)

apolynomial of

degree

r we define

p(z)

z"

,(z-1).

For

nonzerocomplexnumberst,

ct"

denotes

1/.

Finally,z denotes eieinthe integrals presented.

2.

A

PAIR

OF

BIORTHOGONAL POLYNOMIAL SETS

In

this sectionwe consider apairofpolynomialsetswhicharebiorthogonalon the unit circle withrespecttothe measure

dv(O).z-"(z_ctl)(z_%)...(z_ah)dO

z.ei

det

z’/"

z’-I

z"

-=

z 1 m-1

C (Z (Z 0. 1

C/-

-1

{L-2

Ct2

Then ifp._

(z)

isapolynomialofdegreeatmost n 1wehave

xp.(z)p._,(z)[z-’(z

-a)(z

-c)...(z

-a)o

o.

LEMMA

2. Stillassumingthat 0 rn h wedefine

9.(z)

by

(

-<)( ;)...(

-,:),.()-det

Z"

+k

Z"

+h-1

Z"

+h-m

Zk-m-1

Zk-m-2

Z l

+h +h-1 +h-m -m-1 -m-2

et

1

(1 (,1 (ll (ll (ll

4.-,

;.-,-,

;..,-.

?-.-,

?-.-(2.2)

3.

APPLICATION TO

THE

POLYNOMIALS

OF PASTRO

In

thissection we will consider theweight

(qz;q2)(R)(qz-’;q2)

w(z)

(aqz;q)(R)(bqz_;q)(R)

z e

andderive

Pastro’s

biorthogonal polynomials using Theorem 1 above and the same idea behind Ismail’s

[5]

proof

of Ramanujan’s lapl-summation. Namely,wechoose appropriate values forthe where no

%

is zeroand

ct.

1/-.

Thenifp._

x(z)

isapolynomialof

degree

atmostn 1wehave

p.

,(z

,.

(z

[z-(z a,)

(z

c).

.(z

a)]a

o

o

THEOREM

1. Let the polynomialsets

{ap,,(z)}

and

{,(z)}

be defined as in the abovetwo

lemmas.

Assume,

moreover,that for each n, ap.(z)and

,(z)

areof precisedegree n.

(This

is

equivalenttoassumingcertain subdeterminants inequations

(2.1)

and

(2.2)

are

nonzero.)

Then, providedthatfor eachn wehave,

,[z-(z

ct,)

(z

,)...(z a,)]a0

,,

0,

these polynomialsetsarebiorthogonalonthe unit circle withrespecttothemeasure

dv(O)-z-(z-aO(z-ag...(z-a,)dO,

z-e

’

628 R.. RUEDENANN

parametersaandb,andthen use analytic continuationtoget the full result. For thechoice ofa

-q2,

andb

-qZ,

wehave

(qz;q2)(R)(qz-;q2)(R)

w(z)

(q2,

z;q2)(R)(q2,

lz-;q2)(R)

(qz ;q

),

(qz-;q

[(1

-qz)(1

q3z)...(1

-q

2,-lz)]

[(1

-qz-)(1

-q3z-)...(

q2-z-)]

q’(-1)’z-’[(z

q42"

’)

(z

q42"

-a).

.(z

q-a)

(z

q-’)][(z

q (z q

a).

..(z

qZ,-,)]

Notethat thezeros of

w(z)

increaseby factors of

q’

and,moreover, theconjugation

w(z)

merely switchestherolesofrands. Wearenowreadytoapplyourlemmas.

Let

h r+ sandm sinLemma1 andlet

qZ,-a,

_qZ,-cq

q4’-),c

z-q’’-),

.,a ct,/,

Definep.(z) by

Z

Z +4-1

Z

Z4-1 Z4-2

Z 1

+r+4-1 -I -2

1

.(z)

-z)det

(3.1)

a,.

a;:/4 a a,+4 a+4 ct,+, 1

where

(z)

denotes theVandermondedeterminant,ordifferenceproduct,on

{Z,q-(2,-1), q-(2,

a)

,q2

-3,

q2-

1}.

Let

ht

denote the complete symmetricfunction on

{z,q-a’-),q

-a’-3),

q2,-a, q2,-}

andlet j denote the

complete

symmetricfunctionon

{q-a,-X),q-a,-a)

,q2-3,q2,-x}.

Weset

he-

jo-1and h_ j. 0fork>O.

Note

that

h,

zh,

-1

+.h

(3.2)

for all integers k.

By

useofthe Jaeobi-Trudiidentity, equation

(3.4)

in

[8],

wemaywrite

W.(z

aet

h

hi

...h,_l

h,/4

h,/4/1

...h,/,/4

h_

h

...h4_

h

h,/4

...h

h..+

...h

h,/

h,+

...h

h..,+

...h_

h,

h,/l

...h,

Thus

,,(z)-

det

andusing

(3.2)

repeatedly

weget where

h.

h,

h,

/2

...h.

h,

h,

h,

...h,

h._,

h h

.-.

}

p,(z)-A,.,z"

+A

....

F

"-x

+

+A,.

F

+A,.

I I

detl.j,

A

-/,)

J.-,/ J.-,/2

""J.

The

problem

now istoevaluate

A.,t

ingeneral We haver +szeros in ourweightfunctionbut

A.,t

is

only

a

(r

+

1) by (r

+

1)

determinant. We"fill

out"

A.,t

andusetheJacobi-Trudiidentityin reverse.

A

Jl

"L

-2

L

-k -1

L

"L

+1

"L

J-x

Jo

""J,-,

L

_-*/,-2

L

+,-1

""L

"".

J-,

J-,/l

""J-2

J,-t-1

J,

J,,.1

""J,/,-1

That is,

A,.

det

J--r

Nowsetting

o,

-(2a"

I)[0+

1+...+

(s-2)

+(n

-k+s-

l)+(n +s)+(n

+s+

I)

+...

+(n

+r+s

I)],

wefind

A.,

q

’

(

1,

q2,q4

qa,-2,

qa,,

-

/.-i),

q2,, /,),

qa.

i),

qa.

b

(q--1),q42,-s),

...,qZ,-S,

q2,-1)

andstraightforwardbut rather tediouscalculationsyield

A,.,/I

-1

(1-qa"-*))(1-q

z(+’/l))

A,,,,

"q

(1-

qa"

-*

/’-1))

(1

qZ(*

1))"

Thus

sothat

A.,t

.q-1

(q2;q2)

-t

(q2;q2)

-/-I

(q2,q2;q2)

(q2,;q2).

-(qZ’;q2)

_t_

630 R.W. RUEDEMANN

wherea

q2"

andb

qZ,.

At

thispointweknow that forasuitableconstant

c.

V,(z)

c,ko

(aq?;q2)(b;q2)"-k

)

(q:’;q)k (q;q?)._

(q-lz

To

sumthings up, ifwedefine as

Pastro

does,

p,,(z,a,b

i

(aq;q2)’ (b;q2)"-’

(q-z)

-0

(q:,;q2)t

(q

;q,),,

_-,

then

z-ei

for ourparticularchoice ofaand b.

Wecould use

Lemma

2tofindtheothersetofpolynomials required for biorthogonality but, asnotedpreviously,theconjugation oftheweightfunction

w(z) merely

switches theroles ofrand

sandhencethoseofaandbaswell. Thusthepolynomials

q,,(z,a,b):

p,,(z,b,a)

satisfy

z-e

At

thispointwehavethebiorthogonality ofthepolynomialsets

{p,(z)}

and

{q,(z)}.

Westill mustcomputethe valueof

p,,(z)q,,(z)w(z)dO.

In

fact thisposesno greatproblem. Itisfairlyeasytoseethat themonicversionsof thepolynomials inTheorem1,callthem

{W,,(z)}

and

{,(z)},

satisfy

-

%(z)

.(z)

[z-(z

a,)(z

ag...(z

a,)]dO

Now

if we let det

det

(1/&

i

+k-1

I

/I m-1 .-2 ₁

q2S-therighthandsideof thepreviousequation becomesapowerof q timesaquotientofVandermonde determinants. Tobeprecise,letP,(z)andQ,,(z)denote the monieversionsof

Pastro’s

p,(z)and

q,,(z)

respectively. Then

P.()Q.(z)[z"(z-q4-)(z-q-’-)’"(z-qZ’-)(z-qZ’-)]

dO

so

(_l)2’-q

..,+

(I

q2,

2,

2)

(i

q2,

2,

+4)...(i

q2,

2,

7.,)

(-1)’q

(I

q2, )

(I

q2,, /4)...(I q2, 7.,)

2nr,

P"(z)Q"(z)(qz;q2)’(qz-1;q2)’dO

(1

q2"/2"+2))(1-q(2"/2"/4))’"(1-q’/2"/2")

(I

q2

2)

(i

q2

/4)...(i

q2, 2,)

and

P,,(z)q,,(z)(qz;q2),(qz-;q’),

dO"

(q:,;q2),,

q"]

Finally,define

f,d,z;q)

(q2;q:’)(R)(abq-;q2)(R)(qz;q2)(R)(qz-;q

:’)(R)

a q2,. (aq

2;q2)(R) (be/2;q2),,

(qaz

;q2)(R) (qbz-;q:)(R),

We get

2. P"()q"(z)

P(z;q2)dO

"q-(q,q2;q2).

(q2;q2)

(q2;q2),

(qZ,q2;q2),,

][(1

2"/2"/2)(1

2,,/2,./4).

(q2;q2),,

q’][

q q

""(1

(1

q

2)

(1

q2,

/4)...(1

q’

2,)

(q

2;q

-)(R)

(q2,

2

(q2,. 2;q

2)(R) (q2 2;q2)(R)

q

(q2,+2,+2

2,

;q ),,

(q2;q2),,

(l-q2+2’+2)(l-q

2"+2"/4)

(I-q2"/2’/2")I

(abq;q’),,

-

q

qZ,

(q2;q2),,

q where a-

b-The full resultfollows by analyticcontinuation. Actually,twoanalyticcontinuations areneeded: firstwithrespecttotheparameterawithbfixedata

qZ,,

then withrespecttotheparameterb.

4.

MODIFICATION OF MEASURES

BY

LAURENT POLYNOMIALS

In

this sectionwestartwithameasure

dv(O)

which isnotnecessarily positiveonz e

.

From Baxter

[2]

weknow that ifcertainToeplitzdeterminantsarenonzero then thereexists aunique pair of polynomialsets which arebiorthogonal onthe unit circle. Wewillcall this pair

{,,(z)}

and

632 R.W. RUEDEMANN

andthatforeach n,

..(z)o,_(z)dv(O)-

O,_(z)av(0)-

0,

,(z),(z)av(O),,O.

Whatwe want to do is multiply the

complex

measure

dv(O)

by a Laurent polynomial and get determinantformulasfor the newbiorthogonal polynomials,

{ap,(z)}

and

{,(z)},

interms of the oldpolynomials,

{,,(z)}

and

{,(z)}.

Actually,weare goingtorestrictourselvesto twotypes of Laurent polynomials, those of the forms

R(z) z-’G,,(z)

and

R

l(z)

z-("

1)G

(z),

where

G,(z)

and

Gz,,

(z)

arepolynomials having precise

degrees

2m and 2m+1respectively. Further-more,weshallrequirethat neither

G,,(z)

or

G

l(z)

havezas afactor. We havetwoeases: the evencaseandthe odd case.

THEOREM

2.

(even case)

Let

{V.(z)}

begivenby

’()

’()

...’-’’()

<)

)

..."

1

(z.(z)-t

(4.)

’)

’<)...-’<)

)

<)...<))

where thezerosof

G(z)

are

{a,

},

(z)

denotes

,

.,(z),

and

(z)

denotes

,

.,(z).

t

{,(z)}

begiven by

z z)

z z)

(z.(z)-t

(.2)

Here

weare assumingthatthezerosof

G(z),

{a,

},

arepaiisedistinct,

ffor

zerosof multiplicity s,s>1,we

replace

thecoesponding rowsin the determinant bythe derivativesof order 0,1, 2, s 1ofthepolynomialsin the firstrow,evaluatedatthat

zero.)

Furthermore,we shallassume that

,(z)

and

,(z)

are bothofprecise

deee

n. isisequivalenttoassumingceain subdeteinants inequations

(4.1)

and

(4.2)

arenonzero.

Thenfor any polynomial p,_

(z)

of

deee

atmostn 1 we have

and, assuming that

then the polynomialsets

{p,,(z)}

and

{,,(z)}

arebiorthogonalontheunit circle withrespecttothe measurez-"Gz,,(z )dv(O).

THEOREM3.(odd ease) Seth 2m + and let

{,,,(z)}

be givenby

G(z)V.(z)

aet

’()

a,’(a,)

"^"

.."

(4.3)

wherethezerosof

Gh(z)

are

{cq,

ct2

ct,},

(z)

denotes(I),,/,,,(z), and

6(z)

denotes

,,

/,,,(z).

Let

((o)

{,,(z)}

begiven by

(4.4)

zG(z

)p,(z

det

Unlikethepreviousthree determinants in

(4.1)

to

(4.3),

here in

(4.4)

(z)

denotes

,,

/,,/l(z),and

(z)

denotes

l(z).

Weare assumingthat thezerosof

zGh(z),

{0,ch,(h

ct,},

arepairwise

distinct.

(We

takecareofzerosofmultiplicitys,s>1as

usual.)

Furthermore,weshallassume that xp,,(z)and

,,(z)

areboth ofprecisedegreen. Thisisequivalenttoassumingcertainsubdeterminants inequations

(4.3)

and

(4.4)

are nonzero.

Then forany polynomial p,,_

l(z)

ofdegreeat most n 1 wehave

f

V.(z)p._,(z)z’/)G.

(z)dv(O)-

f

ip.(z)p._(z)z

’/)r-

..

/(z)a(o)-

o

and, assumingthat

)G

.(z).(z)z

.(zv(O)O

then thepolynomialsets

{,(z)}

and

{.(z)}

arebiorthogonalon theunit circle withrespecttothe measure z

"

")G.

(zv(O).

e

unusualform of the determinant in

(4.4)

comesaboutas we are

writing

zG*

.

in the form z

"

zG

dz)]

sothat thesameidea behindTheorem 2appliesina sense.

5.

PROOFS

634 R.W. RUEDEMANN

xl,.(z)p._,(z)[z’(z -ct,)(z

-)...(z -ct)]ao

o.

However,

multiplyingbothsidesof

(2.1)

by

z-"

and thenexpandingthe determinantalongthefirst

row we find that

+h-m

.(z)[z’(z-ct)Cz-c)...Cz-ct)]-

E

c

z

+

E

c,,z

-1

As

each of thetermsin the above sums areorthogonaltoany p,,

(z)

the resultfollowsimmediately.

PROOF

OF

(2.2).

Wewant toshow that if

,,(z)

is definedasinequation

(2.2)

thenforany polynomial p,_

x(z)

of degreeat most n wehave

f

p._,(z) .(z)

[z-’Cz ct,)

(z

a)...(z

ct)]ao

o.

Conjugatingboth sidesofthisequationwe seeitisequivalenttoshowingthat

,c-

o._,Cz

,.cztz-"--’c

-,)c -)...Cz -,’)o o

and

(2.2)

followsas an instanceof

(2.1)

with mreplaced byh-mandthe

at’s

replaced

by

ct,’s.

PROOF OF (4.1). We

wanttoshow thatifap,,(z)is defined as inequation

(4.1)

then for any polynomial p,,_

(z)

of degreeat mostn 1 wehave

,,(z)

p._

,(z)

z’G,Cz)av

(0)

0.

We onlyhavetwotypesofpolynomialsin the first rowof the determinant in

(4.1).

Weconsider

eachseparately.

Let

p,,_

(z)

beany polynomial of degreeat most n 1.

(i)

Then, for the polynomials

zl,,

/,,,(z),

where 0,1,2 m,wehave

"z’

’z"

(z

p,,_(z)dv(O)-

,,/,,,(z)z’-’p,,_a(z)dv(O)-O.

(ii)

Forthepolynomials

zt:

/,(z)

wehave

z"

p,,_(z)dv(O)-

zt-"z"/’,,,/,,,(1/z)p,,_(1/z)dv(O)

I

z’/

,.,/.(z)z

"-

p._

(llz)av(O)

[...(z)[z’"o;_,Cz)lv(O

-0

but thisissimply

.(z

p.

(z

z-"G(z

)dv(0) 0.

[Note:

for thepolynomials

zt

/,,(z)

wewill usethe choice --1 inour

proof

of

(4.3).

Also,in

(4.1)

wemayallow 0 values for some

%’s.

In

fact,wemake useofthis choicein

(4.4).

Theonly reason wearerestrictingtheir values heretobe nonzero isbecause ofthe

et.’s

in

(4.2).]

PROOF OF (4.2).

For

,,(z)

asdefinedby

(4.2)

wewant toshow that

.Cz)

p._,Cz)z-’C,.Cz)av(0)

o.

Equivalently,we wishtoshow that

f

Zv.(z)

p._

(z)

z-’.(z)

av(O)

o

and this we getsimply by applying

(4.1)

tothe modification of the measuredv(O) bytheLaurent polynomial

z-’G,(z).

Note

thathaving

dr(O)

rather than

dr(O)

simplyswitchesthe roles of

z)

and

z)in

(4.1).

PROOF

OF

(4.3).

Wewanttoshow that ifap,,(z)isdefined as inequation

(4.3)

then

.(z)

p._(z)z"

"a.

(z)av(O)

O

Considering thefirstrowofthe determinant in

(4.3)

weseethat this isequivalenttoshowingthat z./

o._(z)av(O)-,./.(z)

z"

p._(z)av(O)-O for 1,2 m +1andthat

p._l(Z)

dv(O)

O

for -0,1,2, m.

However,

these statements areequivalentto

(i)

and

(ii)

in our

proof

of

(4.1).

PROOF OF (4.4).

Finally,wewant toshow that if

.(z)

is defined as inequation

(4.4)

then

.(z) o.-

(z)z

"

".

,(zv(O)

o

thatis, we want

Herethe

problem

isthat

.(z)Co._(z))(z"

.

(z))(av(o))

o

z

-’/’)Gz.+

,(z)

(constant)

z4"

/’)G,/

l(z)

sothatwe cannot use

(4.3)

toget

(4.4).

In

fact,

636 R.W. RUEDEMANN

However,

wegetaround thisby applying

(4.1)

tothe modificationof

dr(O)

byz4’’

)H2t,,,.

(z)where

U:,,,

/,,(z):

[This

iswhataccountsfor the unusualform of

(4.4).]

6.

REMARKS

Wehavefoundwemay modifythe

Lebesgue

measure dO ontheunitcircleby multiplication by any Laurent polynomial whose zerosweknow, providedcertain determinantswere nonzero. Whenwepassedtothemoregeneral

problem,

as wedidinsection4, ofmultiplyinganunknown

measuredv(O) by Laurent polynomials,werestrictedwhich

Laurent

polynomialswecould use.This

made theproofsfor thatsectionstraightforward. However,thisrestrictionisunsatisfying--atleast

totheauthor--butatthepresenttime it is stillunresolved.

ACKNOWLEDGMENT

The author thanksDr.M. E. H.Ismail for making this work possible.

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