inverse finite-type relations between sequences of

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el case en que solo la sucesi6n {Bn }n2>:O es ortogonal. De hecho, encontramos condiciones necesarias y suficientes que conducen a relaciones de tipo finito ...

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INVERSE FINITE-TYPE RELATIONS BETWEEN SEQUENCES OF POLYNOMIALS By

Francisco Marcellan I & Ridha SfaxP Abstract Marcellan, F. & R. Sfaxi: Inverse finite-type relations between sequences of polynomials. Rev. Acad. Colombo Cienc. 32( 123): 245-255, 2008. ISSN 0370-3908.

Let be a monic polynomial, with deg = t 2: O. We say that there is a finite-type relation between two monic polynomial sequences {Bn }n~O and {Qn }n~O with respect to , if there exists (s,r) E N2 , r 2: s, such that n+t

(X)Qn(X)

=

2:::

An,..,B..,(x), n 2: s, with Ar,r-s =I- O.

(*)

v=n-s

The corresponding inverse finite-type relation of (*) consists in a finite-type relation as follows: n+s

n:(x;n)Bn(x)

=

2:::

e~,..,Qv(x), n 2: t, with e;+t,r =I- 0,

v=n-t

=

where degn;(x;n) s, n 2: t. When the orthogonality of the two previous sequences is assumed, the inverse finite-type relation is always possible [11]. This work essentially studies the case when only the sequence {Bn}n~o is orthogonal. In fact, we find necessary and sufficient conditions leading to inverse finite-type relations. In particular, the structure relation characterizing semi-classical sequences is a special case of the general situation. Some examples will be analyzed.

2

Departamento de Matematicas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain. Correo electr6nico: [email protected] Departement des Methodes Quantitatives, Institut Superieur de Gestion de Gabes, Avenue lilani Habib 6002, Gabes, Tunisie. Correo electr6nico: [email protected] 2000 Mathematics Subject Classification: 42C05, 33C45.

REV. ACAD. COLOMBo CIENC.: VOLUMEN XXXII. NUMERO 123-JUNIO DE 2008

246

Key words: Finite-type relations, recurrence relations, orthogonal polynomials, semiclassical polynomials.

Resumen Sea r./> un polinomio moniea, con deg q) = t 2: O. Decimos que hay relacion de tipo finite entre dos sucesiones de polinomios monicos {Bn }n;::o Y {Qn }n2':O con respecto a :O es ortogonal. De hecho, encontramos condiciones necesarias y suficientes que conducen a relaciones de tipo finito inversas. En particular, la la relacion de estructura que caracteriza a las sucesiones semiclasicas es un caso especial de la situacion general. Se estudian varies ejemplos.

Palabras clave: Relaciones de tipe finito, relaciones de recurrencia, polinomios ortogonales, polinomios semi clcisices.

1. Introduction and background Let JP' be the linear space of complex polynomials in one variable and JP" its topological dual space. We denote by (u,!) the action of u E II''' on j E JP' and by (u}n:= (u,x n ), n 2: 0, the moments ofu with respect to the polynomial sequence {xn }n>O. We will introduce some useful operations in JP'. For any linear funct.ional u and any polynomial h, let Du = u' and hu be the linear functionals defined by duality

(u',f):= - (u,f'), (hu, f) :=(u, hf),

j E JP',

n,

Let recall the following results [11], Lemma 1.1. For any u E ll" and any integer m 2: 1, the following statements are equivalent, (u, Bn) = 0, n 2: rn,

)..m-I

of 0,

As a consequence, the dual sequence {u~1 }n>O of the sequence {B~II}n"O' where B),!](x} = (n+ l)-lB~+l(X), n 2: 0, satisfies

Definition 1.2. The linear functional u is said to be regular if there exists a monic polynomial sequence {Bn}n>o such that

(u, BnBml = bnon,m,

j, h E JP'.

Let {Bn}n>o be a monic polynomial sequence (MPS), degBn = n 2: 0, and {un}n>O its dual sequence, Un E JP", n ::> 0, defined by (un, Bm) := On,m' n, m::> 0, where on,m is the Kronecker symbol.

i) (u, Bm-I) of 0,

ii) There exist)... E C, 0 o satisfies the three-term recurrence relation

B n+2 (x) = (x - ;3n+!)Bn+1(x) - "in+1Bn(X), n 2: 0, B , (x) = x - ;30, Bo(x) = 1, (1.5) where "in+!

# 0, n 2:

°

(see [4]).

(n,l/,m) E 1\1"'

(1.6)

In particular, one has

bV _ n,m -

{o,(bn/b

if 1/ + m < n, if v = n - rn,

m ),

o :s: m

< n,

m

=" n.

°="

v ;::: 0,

Let q, be a monic polynomial, with deg q, = t 2: 0, For any MPS {Bn }n2-0 and {Qn}n2-0 with dual sequences {un}n>O and {vn }n>O respectively, the following formula always holds

n+t q,(x)Qn(x) = L An,vBv(x), n 2: 0,

(1.7)

is

" )n+' (0 n,v v=n-t. n > _ t,

a

MPS, (1.11)

a system of complex numbers (SCN), with O~,n+' = 1, n

In the sequel and under the assumption of the previous definition, we need to put b~m = b:;;,'(u,xVBmBn),

O;+t,r # 0, where {n;(x; n)}n2-t degn;(x; n) = s, n 2: t, and

247

2: t,

When both two sequences are orthogonal, the inverse relation is always possible, In this case, the polynomials n;(x; n), n 2: 0, are independent of n, (see [12], Proposition 2.4), As a current example, we can mention the two structure relations characterizing the classical polynomials, (Hermite, Laguerre, Bessel, Jacobi, see [11]), which could solely be two inverse finite-type relations, In other studies, we find several situations where one of the two sequences is orthogonal. For example, the structure relations characterizing semi-classical sequences associated with Hahll's operators Lq,w, with parameters q and w, [9], The Coherent pairs and Diagonal sequences are also examples of finite type-relations [7,12,13,14], But the inverse relations corresponding to other finite-type relations are not yet considered,

",=0

where An,v = (uv, q,Qn) ,

°="

1/

=" n + t,

n 2: 0.

Definition 1.3, ([12]) If there exists an integer s 2: such that

q,(x)Qn{.'r)

n+t L An,vBv(x), n 2: s,

=

°

(1.8)

v=n-s

and

:3 r 2:

S,

A1' ,1'-8

=f=.

0,

(1.9)

then, we shall say that (1.8) - (1.9) gives a finite-type relation between {Bn }n2-0 and {Qn}n2-0, with respect to lP, When instead of (1. 9), we take An,n-s

i=- 0, n 2:

s,

(1.9')

we shall say that (1.8) - (1.9') is a strictly finite-type relation. The corresponding inverse finite-type relation of (1.8) - (1.9) consists in establishing, whenever it is possible, a finite-type relation between {Qn}n>O and {B n }n2-0, as follows

n;(x;n)Bn(x) =

n+. L O~vQv(x), n 2: t, v=n-t

The paper essentially gives a necessary and sufficient condition allowing the existence of the inverse finitetype relations when the orthogonality of the sequence {Bn }n2>:O is assumed. From now on, it would be necessary to study the case where the sequence {Qn}n>O is orthogonal. It would be very useful to deal with many other situations like General Coherent pairs, see [6,8] in the framework of Sobolev inner products,

(1.10)

2, A basic result

'V'wre use this section to introduce some auxiliary result for the proof of the main theorem in section 3. Lemma 2,1. Suppose {Bn}n>o is a MOPS and {Qn}n>O fulfils (1.8) - (1.9), where t = and s 2: 1. For any SCN (On,v)~;t~, n 2: 0, where On,n+, = 1, n 2: 0, and Br,r # 0, there exist a unique MPS {n,(x; n)}n2-0, degn,(x;n) = s, n 2: 0, and a SCN (d~L)~;;;~_" n 2: 0, such that

°

n+s

n+s

LOn,vQv(X)

=

O~liBi(x)

L i=n-s

= n,(x; n)Bn(x)

+

n-1 L (~LBv(x), n 2: 0, (2,1) v=n-s

REV. ACAD. COLOMBo CIENC.: VOLUMEN XXXII. NUMERO 12]·JUNIO DE 2008

248

Multiplying by Bm(x) and using the orthogonality of

where

{Bnk::o,

min(n,i)+s

L

O~li =

On,vAv,i, n-8::; i::; n+8,n:': 0,

(2.2)

v=max(n,i)

[01

Br .r -

s

= Br.rAr,r-s i= 0,

(2.3)

0n,vAv,m

b;:;,I (u, n,(x; n)BnBm) + (~~\n,

=

lI=n

n - 8::; m::; n -l,n:': 0, (2.4) n+,

L On,vAv,m = b;;,1(u, n,(x; n)BnBm),

v=m

n ::; m ::; n +

8 -

1, n:': O.

(2.5)

Proof. Let (On,v)~;!;~, n :': 0, where On,n+s = 1, n :': 0, and (Jr,r # 0, be a SCN. From (1.8) - (1.9), with t = 0 and 8 :': 1, we get n+s

n+s

v

v=n

v=n

i=v-s

n+s

n+s

L (In,vQv(X) = L (In,v L =

L (In,v L

v=n

=

L

(J~li8m,i = b;;,1(u, n,(x; n)BnBm) +

L d~L8m,v.

i=n-s

Finally, for n ::; m ::; n (2.5).

+8

1 and n :': 0, we deduce

-

0

Proposition 2,2, Assume {Bn}n>o is a MOPS and {Qn}n~O fulfils (1.8) - (1.9), with t :': 1. For any SCN (Bn,v)~:!~_t' n 2: 0, where Bn,n+s = 1, n :2: 0 a.nd (Jr+t,r # 0, there exist a unique MPS {n,+t(x; 11)}n~O, where degn,+t(x; n) = 8 + t, 11 :': 0, and a SCN (~!v)~;;;~-,-t' n:': 0, such that for every integer n:': 0 n+s

q,(x)

L

n+s+t

On,vQv(x)

=

v=n-t

L

(J~I,Bi(X)

i=n-s-t

Av"Bi(x) n-I

+

Xi,vAv" Bi(X), n:': 0,

L

(~,tBv(x),

(2.7)

v=n-s-t

i=n-s

where, for each pair of integers (i,l/) such that n -. i ~ n + s and n ~ v ~ n + s, we took Xi,v

n-I

In particular, for 0 ::; m ::; n - 8 - 1 and n :': 8 + I, it follows that d~'m = O. Hence, (2.1) holds. Moreover, for 11 - 8 ::; m ::; 11 - 1 and n :': s, we recover (2.4).

m+,

L

n+s

where 8 ::;

min(n,i)+s+t

B~~i

if v - s :S i :S v,

I, 0, {

=

L

(}n,v-tAv-t,i ,

v=max(n,i)

otherwise.

11 -

s - t ::; i ::; n

+ s + t,

(2.8)

The permutation of these two sums yields n+s

n+s

v=n

i=n-s

L (In,vQv(X) = L

O~,liB,(x),

m+s+t

L

where min(i+s,n+s)

L

B~~li

b;:;,1(u, n,+t(x; n)BnBm) + (~!m' n - s - t ::; m ::; n - 1, (2.10)

n+s+t

n - s S i :-::; n + s, n =

=

Bn ,II Av,i,

v=max(n,i)

(J~~~_,

(In.v-tAv-t,m

(Jr,rAr,r-,

~

0,

(2.6)

L

(In.v-tAv-t,m = b;;,l(u,n,+t(x;n)BnB m ),

l.I=m

# O.

n ::; m ::;

11

+ s + t - 1. (2.11)

Hence, (2.2) and (2.3) are valid. The Euclidean division by Bn(x) in the right hand side in (2.6) gives n+s

L

i=n-8

n-I

(J~I,B,(x) = n,(x; n)Bn(x)+

L

v=O

(~~LBv(x), n:': O.

Proof The case t = 0 was analyzed in Lemma 2.1. Let us take t:': 1. Consider the MPS {Pn}n~t defined by (2.12)

MARCELLAN. F. & R. SFAXI: INVERSE FINITE-TYPE RELATIONS BETWEEN SEQUENCES OF POLYNOMIALS

From (1.8) - (1.9). we have

3. A matrix approach and main results

n

Pn(X)

L

=

In this section, we will work under the assumptions of the Proposition 2.2 and we will give a matrix approach to our problem.

)..n.vBv(x). n :;0: t.

v=n-t-s

An-t,,,,) n - t - s :S v :::; n, n 2: t, and =f. O. Now, let (Bn,v)~;:~_t' n 2: 0, where 1. n :;0: 0, and Br+t.r oF 0. be a SCN. One

where ).n,v 'xt+r,r-s

Bn .n +, =

has

n+,

L

. n+s,n+s+f.-l )T . -

Our data are en, En, Fn , M n , Sn, T n , Kn and our unknowns are Vn and Wn .

From (3.2), we get Vn = S;;'(M n 8 n

-

i) t. n (t,8),. 0, n 2: p. ii) There exist a unique SCN (O~.vl~!~_t, n 2: p,

Fn ).(3.3)

Thus, substituting in (3.1) we get Kn8n - Wn - En = T nS;;'(M n n - Fn ), i.e,

e

Wn = (Kn - TnS;;lMn)8n + T n S;;' Fn - En. As a consequence, for every choice of From (3.3), we deduce Vn .

en!

Theorem 3.2. Let {Bn}n>o be a MOPS and {Qn}n>O be the MPS satisfying (1.8) - (1.9). For each fixed integer p 2: t + 1, if we suppose that p, degn;(x; n) = s, n2: p, such that

p

n+,

L

n;(x; n)Bn(x) =

(~,vQv(x), n 2: p.

(3.8)

v=n-l

F(Qn, ... , Qn-I) = G(Bn, ... , B n -,),

where F and G are fixed functions. When F and G are linear functions, some situations dealing with the inverse problem have been analyzed in [1,2J. There, necessary and sufficient conditions in order to {Qn}n~O be orthogonal are obtained. This kind of linear relations reads as follows. There exist.s (l,s,r) E l\I3, with r 2: such that n

L

s=

Proof. From Theorem 3.2, with t = 0, there exists the corresponding inverse finite-type relation associated with the relation (3.7) if and only if tJ.n(O, s) i' 0, n 2: p. Equivalently, there exist a unique SCN (e~,v)~~~' n 2: p, where B~,n+s = 1, n 2:: p, and B;,1' 1= 0, if p :S: T, and a unique MPS {n;(x;n)}n2:p, degn;(x;n) = s, n 2: p, such that n+,

max(l,s)

L

n;(x;n)Bn(x) =

n

(n,vQv(x) =

L

>'n,vBv(x), n 2: .;,

(3.6)

v=n-s

v=n-l

with (r,r-I>'"r-,

i' O.

Here, (n,n

+ SnPn-1 (x) = R~I}(X) + tnR~~1 (x), where tn i' 0, for every n 2: 1, and with the condition tl i' SI· Pn(x)

n 2: 1,

n:(x;n)Bn(x) =

v=n

i=v-/

L

n 2:

s.

n+s

L

e~,v

Xi,v(v,i Qi(X), n 2: p,

i=n-/

where, for each pair of integers (i, v) sHch that n - I S + sand n ::; v ::; n + s, we took

_ {1,

0,

Xi,v =

if, v - l ::; i

~

v,

otherwise,

The permutation inside these two sums yields n+s

L

n:(x; n)Bn(x) =

(~,iQi(X),

i=n-l

where

n

>'n,vBv(x),

(v,iQi(X)

i ::; n

the corresponding inverse finite-type relation between two sequences satisfying (3.6) is possible under certain conditions. Indeed, let consider the MPS {Gn}n>, given by

v

L e~,v L

v=n

technical

In the same context of our contribution, we show that

n+s

n+s

=

Notice that in general {R~I}n2:0 is not. a MOPS.

L

(3.9)

But from (3.6) and (3.7), the above expreRSion becomes

= >'n,n = 1, n 2: s.

More recently, in [5], A, M_ Delgado and F. Marcelhin exhaustively describe all the set of pairs of quasidefinite (regular) linear functionals such that their corresponding sequences of monic polynomials {Pn}n>O and {Rn}n~o are related by a differential expression

Gn(x) =

e~,vGv(x), n 2: p.

v=n

min(i+l,n+s)

(~,i

(3.7)

v=n-s

=

L

(J~,v(v,i

,

v=max(n,i)

With the finite-type relation between the sequences {Gn }n2:' and {Bn }n2:', we can associate the determinants tJ.n(O, s), n 2: s. So, we have.

if n - I S i S'n

Corollary 3.3. Let {Bn}n2:0 be a MOPS and {Qn}n2:0 be the MPS satisfying (3.6). For each fixed integer

ifpSr.

+ s, n 2: p, and

o

252

REV. ACAD. COLOMBo CIENC VOLUMEN XXXII, NUMERO 12J-JUNIO DE 2008

4. The case: (t,s) = (0,1)

where ,

"In,

= -,-- + An+l,n

vn,o

Let {Bn}n>o be a MOPS with respect to the linear functional u and satisfying the three-term recurrence relation (1.5). Consider the following finite-type relation between and {Qn}n~O, with index s = 1, with respect to q,(x) = 1, {Bn}n~o

(4.1) ~r

2: 1,

Ar ,r-1 =F 0.

(4.2)

From Lemma 2.1, for every set of complex numbers, On,n, n 2: 0, with Or,r =F 0, there exists a unique MPS {111 (x; n)}n>O, where 111 (x; n) = x + vn,o, n 2: 0, and a

unique set of complex numbers, Qn+I(X)

+ On,nQn(x)

(~oJn_ll n 2: 0, such that

111 (x; n)Bn(x)+

=

(~ln_IBn-l(x), n 2: 0,

(4.3)

where An,n-IOn,n

{ fJn,n - vn,o

_

The determinants associated with (4.1) - (4.2) are given by

Ll. 0 (0,1) = 0,

Ll. n (O,l) = An,n-I, n 2: 1,

(4.5)

where Ll.r(O, 1) = Ar,r-I =F 0. As a consequence of Theorem 3.2, when t = and s = 1, we have the following result

°

Proposition 4.1. Let {Bn}n>o be a MOPS and {Qn}n>O be the MPS satisijring (4.1) - (4.2). For every fixed integer p 2: 1, the following statements are equivalent i) An,n-I =F 0, n 2: p. ii) There exist a. unique set of complex numbers O~,n =F 0, n 2: p, and a unique MPS {l1i(x;n)}n~p, degl1j(x;n) = 1, n 2: p, such

the functional equation u' + 1/Ju = 0, (4.9) 2 where 1/J(x) = -ix + 2x - i(a - 1) and with regularity condition a 'tc Un>oEn, where Eo = {a E o be a MOPS anG {Qn}n20 be tlle MPS satisfying- (5.1) - (5.2). For ev· ery fixed integer p 2: 2, the following statements an equivalent

(5.5)

and s = 2,

i) An ,n-2(A n +l,n-1 - "In) of 0, n 2: p.

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254

ii) There exist a unique SCN (B~.v)~i~, 71. ~ p, with B~,n+2 = 1) n ;: : p, and e;,r =I- 0, if p :S r, and there exists a unique MPS {n2(x; 71.)ln:"p, where degn2(x;71.) = 2, 71. ~ p, such that, for

B~'12(x) + en,nB~I(x)

=

+ (~~~_2Bn-2(X),

n 2(x; 71.)Bn(x) =Qn+2(X) +B~,n+! Qn+! (x) +e~,nQn(x) . (5.6)

= 1'1 + vo,o, = I'n+l + I'n

+ vn,o, 71. ~ 1, (5.10) = I'nl'n-l + (~~~-2' 71. ~ 2.

[A n ,n-2(tJn-l-{3n+l +"\n+2,'1.+l}-).n,'1.-l')'n-l]')'n An ,n-2("\n+l,'1.-l

n,n41 -

0*

=

n,n

(5.9)

where

We write _

0, we

(x 2 + vn,o)Bn(x)

71. ~p,

()*

~

are symmetric, i.e, Bn(-x) = (-l)nB n (x), n get, for n 2: 0,

- ')'n)

Since we have An ,n-21 n 2: 2, the choice (~O~_2 2, is possible and yields the inverse relati~n

/n')'n-l >''1.,n-2 '

=

0, n 2:

n;(x; 71.) = x 2 + V~"X + v~,o, 71. ~ p, (5.7) where

where

+ (A n+l,n - f3n)f)~,n+l -I'n+l ),n+2.n + f3n(f3n+l - ),n+2.n+l), B~,n+l - i3n+l - !3n + An+2,n+l.

v~,o = e~,n

V~,l =

71.+1

=

j

+OO

-00

p(x)e- x ' dx.

This sequence of polynomials was introduced by P. Nevai (see [15]) in the framework of the so-called Freud measures. These polynomials satisfy the three-term recurrence relation (1.5), with coefficients f3n = 0, 71. ~ 0, and where ')'n+l, n 2: 0, are given by a non-linear recur-

rence relation (see [3] and [15])

71. = 41'nbn+l

+ I'n + I'n-l), 71.

~ 1,

with '1'0 = 0 and '1'1 = r(3/4)r(1/4). The sequence {Bn}n>o satisfies the following structure relation ( see [3]) B~I(x) = Bn(x) + ),n,n-2Bn-2(X),

71.

~ 2,

(5.8)

where

),n,n-2

4

= 71.

+ 1 'In+l'lnl'n-l io 0,

71.+1

--, 4')'n+l

v~,o = _ - - I'n - 'Yn+l 4 ')'n+l

Example. Let {B n }n2:0 be the sequence of monic polynomials, orthogonal with respect to the linear functional u such that (u,p)

e~,n=

l'n+

71. ~ 2.

From (5.3), with Qn(x) = B~ll(x), 71. ~ 0, and the fact that the polynomial sequences {B n }n2:0 and {B~I }n2:0

4

+ -+ 31'n+1'l'n+2'l'n+3' n

Here, the determinants associated with (5.8) are

4 1 I'n+1'l'nl'n-l 2 [4 b.n(O, 2) = 71. + 71. + 2 I'n+21'n+l - 1] , (5.12) 71. ~ 2, with b.o(O, 2) = b. , (0, 2) = O. From Proposition 5.1, we deduce that the uniqueness of the previous inverse relation requires that

An+1.n-l -

I'n = I'n [n!21'n+21'n+l - 1] io 0, n ~ 2. Equivalently, 41'n+21'n+l io 71.+2, 71. ~ 2. Indeed, by using (5.8), where n is replaced by 71. + 1 and taking into account the orthogonality of the polynomial sequence {Bn}n>o, we get B~L (x) = xBn(x) + (),n+1.n-l - I'n)Bn- l (xl, 71. ~ 1. On the other hand, if we suppose that there exists an integer no 2: 2 such that Ano +l,no-l - "'tno = 0, then B~~+I (x) = xBn" (x). In this case (5.11), with 2 71. = no will be written as (x + ax + v~ o)Bn(x) = I'l [11 • [11 ' . B n+2(x) + aBn+l (x) + en,nBn (x), for all a E