# Inverse finite-type relations between sequences of polynomials.

ResumenSea [fi] un polinomio monico, con deg [fi] = t [flecha diestra] 0. Decimos que hay relacion de tipo finito entre dos sucesiones de polinomios monicos [{[B.sub.n]}n[flecha diestra]0] y [{[Q.sub.n]}n[flecha diestra]0] con respecto a [fi], si existe (s, r) [elemento de] [N.sup.2], r [flecha diestra] s, tal que

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII]

La correspondiente relacion de tipo finito de (*) consiste en una relacion de tipo finito como sigue:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII],

donde deg [[OMEGA].sup.*.sub.s] (x; n) = s. n [flecha diestra] t. Cuando se supone la ortogonalidad de las dos sucesiones previas, la relacion de tipo finito inversa siempre es posible [11]. En este trabajo se estudia el caso en que solo la sucesion [{[B.sub.n]}.sub.n[flecha diestra]0] 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 varios ejemplos.

Palabras clave: Relaciones de tipo finito, relaciones de recurrencia, polinomios ortogonales, polinomios senil clasicos.

Abstract

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

Let [phi] be a monic polynomial, with deg [phi] = t [greater than or equal to] 0. We say that there is a finite-type relation between two monic polynomial sequences ([B.sub.n]}n[greater than or equal to]0] and [{[Q.sub.n]}.sub.n[greater than or equal to]0] with respect to [phi], if there exists (s, r) [member of] [N.sup.2], r [greater than or equal to] s, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where deg [[OMEGA].sup.*.sub.s] (x; n) = s, n [greater than or equal to] 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 [{[B.sub.n]}.sub.n[greater than or equal to]0] 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.

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

1. Introduction and background

Let P be the linear space of complex polynomials in one variable and P' its topological dual space. We denote by (u, f) the action of u [member of] P' on f [member of] and by [(u).sub.n] := (u, [x.sup.n]), n [greater than or equal to] 0, the moments of u with respect to the polynomial sequence [{[x.sup.n]}.sub.n[greater than or equal to]0].

We will introduce some useful operations in P'. For any linear functional u and any polynomial h, let Du = u' and hu be the linear functionals defined by duality

(u', f) := - (u, f'), f [member of] P, (hu, f) := (u, h f), f, h [member of] P.

Let [{[B.sub.n]}.sub.n[greater than or equal to]0] be a monic polynomial sequence (MPS), deg [B.sub.n] = n, n [greater than or equal to] 0, and [{[u.sub.n]}.sub.n[greater than or equal to]0] its dual sequence, [u.sub.n] [member of] P', n [greater than or equal to] 0, defined by ([u.sub.n],[B.sub.m]) := [[delta].sub.n,m], n, m [greater than or equal to] 0, where [[delta].sub.n,m] is the Kronecker symbol.

Let recall the following results [11].

Lemma 1.1. For any u [member of] P' and any integer m [greater than or equal to] 1, the following statements are equivalent.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

Definition 1.2. The linear functional u is said to be regular if there exists a monic polynomial sequence [{B.sub.n]}.sub.n[greater than or equal to]0] such that

(u, [B.sub.n] [B.sub.m]) = [b.sub.n] [[delta].sub.n,m], n, m [greater than or equal to] 0, (1,2)

where

[b.sub.n] = (u, [B.sup.2.sub.n) [not equal to] 0, n [greater than or equal to] 0. (1.3)

Then the sequence [{[B.sub.n]}.sub.n[greater than or equal to]0] is said to be orthogonal (MOPS) with respect to u.

As a straightforward consequence we get

* The linear functional can be represented by u = [(u).sub.0][u.sub.0], and the following relations hold

[u.sub.n] = [b.sup.-1.sub.n] [B.sub.n] u, n [greater than or equal to] 0. (1.4)

** The sequence [{[B.sub.n]}.sub.n[greater than or equal to]0] satisfies the three-term recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)

In the sequel and under the assumption of the previous definition, we need to put

[b.sup.v.sub.n,m] = [b.sup.-1.sub.m] (u, [x.sup.v] [B.sub.m] [B.sub.n]], (n, v, m) [member of] [N.sup.3]. (1.6)

In particular, one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [phi] be a monic polynomial, with deg [phi] = t [greater than or equal to] 0. For any MPS [{[B.sub.n]}.sub.n[greater than or equal to]0] and [{[Q.sub.n]}.sub.n[greater than or equal to]0] with dual sequences [{[u.sub.n]}.sub.n[greater than or equal to]0] and [{[u.sub.n]}.sub.n[greater than or equal to]0] respectively, the following formula always holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

Definition 1.3. ([12]) If there exists ah integer s [greater than or equal to] 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

and

[there exists] r [greater than or equal to] s, [[lambda].sub.r,r-s] [not equal to] 0, (1.9)

then, we shall say that (1.8)-(1.9) gives a finite-type relation between [{[B.sub.n]}.sub.n[greater than or equal to]0] and [{[Q.sub.n]}.sub.n[greater than or equal to]0], with respect to [phi].

When instead of (1.9), we take

[[lambda].sub.n,n-s] [not equal to] 0, n [greater than or equal to] 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 [{[Q.sub.n]}.sub.n[greater than or equal to]0] and [{[B.sub.n]}.sub.n[greater than or equal to]0], as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)

[[theta].sup.*.sub.r+t,r] [not equal to] 0, where [{[[OMEGA].sup.*.sub.s] (x; n)}.sub.n[greater than or equal to]t] is a MPS, deg and [[OMEGA].sup.*.sub.s] (x; n) = s, n [greater than or equal to] t, and

[([[theta].sup.*.sub.n,v]).sup.n+s.sub.v=n-t], n [greater than or equal to] t, (1.11)

a system of complex numbers (SCN), with [[theta].sup.*.sub.n,n+s] = 1, n [greater than or equal to] t.

When both two sequences are orthogonal, the inverse relation is always possible. In this case, the polynomials [[OMEGA].sup.*.sub.s] (x; n), n [greater than or equal to] 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 Hahn's operators [L.sub.q,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.

The paper essentially gives a necessary and sufficient condition allowing the existence of the inverse finite-type relations when the orthogonality of the sequence [{[B.sub.n]}.sub.n[greater than or equal to]0] is assumed. From now on, it would be necessary to study the case where the sequence [{[Q.sub.n]}.sub.n[greater than or equal to]0] 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.

2. A basic result

We use this section to introduce some auxiliary result for the proof of the main theorem in section 3.

Lemma 2.1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The permutation of these two sums yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.6)

Hence, (2.2) and (2.3) are valid.

The Euclidean division by [B.sub.n](x) in the right hand side in (2.6) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular, for 0 [less than or equal to] m [less than or equal to] n - s - 1 and n [greater than or equal to] s + 1, it follows thats [[xi].sup.[0].sub.n,m] = 0. Hence, (2.1) holds. Moreover, for n - s [less than or equal to] m [less than or equal to] n- 1 and n [greater than or equal to] s, we recover (2.4).

Finally, for n [less than or equal to] m [less than or equal to] n + s- 1 and n [greater than or equal to] 0, we deduce (2.5).

Proposition 2.2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.7)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.11)

Proof. The case t = 0 was analyzed in Lemma 2.1. Let us take t [greater than or equal to] 1. Consider the MPS [{[P.sub.n]}.suib.n[greater than or equal to]t] defined by

[P.sub.n+1] (x) = [phi](x)[Q.sub.n](x), n [greater than or equal to] 0, (2.12)

From (1.8)-(1.9), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But from Lemma 2.1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.14)

for every integer n [greater than or equal to] 0, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, by using (2.13), (2.14), and taking into account the expressions of [[??].sub.n,v] and [[??].sub.n,v], we find the desired results.

3. A matrix approach and main results

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the same way, using [[theta].sub.n,n+s] = 1, (2.11) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every n [less than or equal to] m [less than or equal to] n + s + t- 1. Replacing m by i + n - 1, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where for i, j = 1, 2, ..., s + t,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, we can use the matrix representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our data are [[THETA].sub.n], [E.sub.n], [F.sub.n], [M.sub.n], [S.sub.n], [T.sub.n], [K.sub.n] and our unknowns are [V.sub.n] and [W.sub.n].

From (3.2), we get

[V.sub.n] = [S.sup.-1.sub.n] ([M.sub.n][[THETA].sub.n] - [F.sub.n]).(3.3)

Thus, substituting in (3.1) we get [K.sub.n][[THETA].sub.n] - [W.sub.n] - [E.sub.n] = [T.sub.n][S.sup.-1.sub.n] ([M.sub.n][[THETA].sub.n] - [F.sub.n]), i.e,

[W.sub.n] = ([K.sub.n] - [T.sub.n] [S.sup.-1.sub.n][M.sub.n])[[THETA].sub.n] + [T.sub.n] [S.sup.-1.sub.n] [F.sub.n] - [E.sub.n].

As a consequence, for every choice of [[THETA].sub.n], we get [W.sub.n]. From (3.3), we deduce [V.sub.n].

On the other hand, there exists a one-to-one correspondence between the vectors [W.sub.n] and [[THETA].sub.n] if and only if the matrix of dimension s + t, [K.sub.n] - [T.sub.n] [S.sup.-1.sub.n] [M.sub.n], is nonsingular.

Under such a condition, there exists a unique choice for [[THETA].sub.n] such that [W.sub.n] = 0. Thus, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Let introduce

[[DELTA].sub.n](t, s) = det([K.sub.n] - [T.sub.n][S.sup.-1.sub.n][M.sub.n]), n [greater than or equal to] 0.

Thus, we have proved the following result

Proposition 3.1. Assume {[B.sub.n]} n [greater than or equal to] 0 is a MOPS and {[Q.sub.n]}n [greater than or equal to] 0 fulfils (1.8)-(1.9). For a fixed integer p [greater than or equal to] t + 1, the following statements are equivalent.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.4)

for n [greater than or equal to] p.

Our main result is

Theorem 3.2. Let {[B.sub.n]}.sub.n[greater than or equal to] 0 be a MOPS and {[Q.sub.n]}n[greater than or equal to] 0 be the MPS satisfying (1.8) - (1.9). For each fixed integer p [greater than or equal to] t + 1, if we suppose that [phi](x) and [B.sub.n](x) are coprime for every n [greater than or equal to] p, then the following statements are equivalent.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

Proof. Taking into account [phi](x) and [B.sub.n](x) are coprime for every n [greater than or equal to] p, from (3.4) we deduce that [phi] divides [[OMEGA].sup.*.sub.s+t] (x; n), n [greater than or equal to] p. So, [[OMEGA].sup.*.sub.s+t](x; ) = [phi](x)[[OMEGA].sup.*.sub.s] (x; n), n [greater than or equal to] p. Hence, the desired result follows.

The orthogonal polynomial sequence {[B.sub.n]}n[greater than or equal to] 0 and the polynomial sequence {[Q.sub.n]}n[greater than or equal to] 0 can be related by a general finite-type relation (see [1]). It reads as follows

F([Q.sub.n], ..., [Q.sub.n-l]) = G([B.sub.n], ..., [B.sub.n-s]),

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 [1,2]. There, necessary and sufficient conditions in order to {[Q.sub.n]}n[greater than or equal to] 0 be orthogonal are obtained.

This kind of linear relations reads as follows.

There exists (l, s, r) [member of] [N.sup.3], with r [greater than or equal to] [??] = max(l, s) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

More recently, in [5], A. M. Delgado and F. Marcellan exhaustively describe all the set of pairs of quasi-definite (regular) linear functionals such that their corresponding sequences of monic polynomials {[P.sub.n]}n[greater than or equal to] 0 and {[R.sub.n]}n[greater than or equal to] 0 are related by a differential expression

[P.sub.n](x) + [s.sub.n][P.sub.n-1] (x) = [R.sup.[1].sub.n] (x) + [t.sub.n][R.sup.[1].sub.n-1] (x), n [greater than or equal to] 1,

where [t.sub.n] # 0, for every n [greater than or equal to] 1, and with the technical condition [t.sub.i] [not equal to] [s.sub.i].

Notice that in general {[R.sup.[1].sub.n]}n[greater than or equal to] 0 is not a MOPS.

In the same context of our contribution, we show that the corresponding inverse finite-type relation between two sequences satisfying (3.6) is possible under certain conditions.

Indeed, let consider the MPS [{[C.sub.n]}.sub.n[greater than or equal to] [??]] given by

[C.sub.n](x) = [n.summation over (v=n-s)][[lambda].sub.n,v][B.sub.v](x), n [greater than or equal to] [??] (3.7)

With the finite-type relation between the sequences [{[C.sub.n]}.sub.n[greater than or equal to] [??]] and [{[B.sub.n]} n [greater than or equal to] [??]], we can associate the determinants [[DELTA].sub.n](0, s), n [greater than or equal to] [??]. So, we have.

Corollary 3.3. Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be a MOPS and [{[Q.sub.n]}.sub.n[greater than or equal to] 0 be the MPS satisfying (3.6). For each fixed integer p [greater than or equal to] max(s,l,1), if [[DELTA].sub.n](0,s) [not equal to] 0, n > p, then here exist a unique SCN [([zeta].sup.*.sub.n,v]).sup.n+s.sub.v=n-l], n [greater than or equal to] p, where [[zeta].sup.*.sub.n,n+s] = 1, n [greater than or equal to] p, and [[zeta].sup.*.sub.r,r-1] [not equal to] 0 if p [less than or equal to] r, and a unique MPS {[[OMEGA].sup.*.sub.s](x; n)}.sub.n[greater than or equal to]p], deg [[OMEGA].sup.*.sub.s] (x; n) = s, n > p, such that

[[OMEGA].sup.*.sub.s](x; n) [B.sub.n](x) = [n+s.summation over (v=n-1)] [[zeta].sup.*.sub.n,v][Q.sub.v](x), n [greater than or equal to] p. (3.8)

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 [[DELTA].sub.n] (0, s) [not equal to] 0, n [greater than or equal to] p. Equivalently, there exist a unique SCN [([[theta].sup.*.sub.n,v].sup.n+s.sub.v=n], n [greater than or equal to] p, where [[theta].sup.*.sub.n,n+s] = 1, n [greater than or equal to] p, and [[THETA].sup.*.sub.r,r] [not equal to] 0, if p [less than or equal to] r, and a unique MPS [{[[OMEGA].sup.*.sub.s](x; n).sub.n[greater than or equal to]p], deg [[OMEGA].sup.*.sub.s] (x; n) = s, n [greater than or equal to] p, such that

[[OMEGA].sup.*.sub.s] (x; n)[B.sub.n](x) = [n+s.summation over (v=n)][[theta].sup.*.sub.n,v][C.sub.v](x), n [greater than or equal to] p. (3.9)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

The permutation inside these two sums yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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

Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] 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 [{[B.sub.n]}.sub.n[greater than or equal to] 0] and [{[Q.sub.n]}.sub.n[greater than or equal to] 0], with index s = 1, with respect to [phi](x) = 1,

[Q.sub.n](x) = [B.sub.n](x) + [[lambda].sub.n,n-l][B.sub.n-l](x), n [greater than or equal to] 0, (4.1)

[there exists]r [greater than or equal to] 1, [[lambda].sub.r,r-1] [not equal to] 0. (4.2)

From Lemma 2.1, for every set of complex numbers, [[theta].sub.n,n], n [greater than or equal to] , with [[theta].sub.r,r] [not equal to] 0, there exists a unique MPS [{[[OMEGA].sub.1](x; n)}.sub.n[greater than or equal to]0], where [[OMEGA].sub.1](x; n) = x + [v.sub.n,0], n [greater than or equal to] 0, and a unique set of complex numbers, [[zeta].sup.[0].sub.n,n-1], n [greater than or equal to] 0, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.3) (4.4)

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

[[DELTA].sub.0](0,1) = 0, [[DELTA].sub.n](0,1) = [[lambda].sub.n,n-1], n [greater than or equal to] 1, (4.5)

where [[DELTA].sub.r](0, 1) = [[lambda].sub.r,r-1] [not equal to] 0. As a consequence of Theorem 3.2, when t = 0 and s = 1, we have the following result

Proposition 4.1. Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be a MOPS and [{[Q.sub.n]}.sub.n[greater than or equal to] 0] be the MPS satisfying (4.1)-(4.2). For every fixed integer p [greater than or equal to] 1, the following statements are equivalent

i) [[lambda].sub.n,n-1] [not equal to] 0, n [greater than or equal to] p.

ii) There exist a unique set of complex numbers [[theta].sup.*.sub.n,n] [not equal to] 0 , n [greater than or equal to] p, and a unique MPS [{[[OMEGA].sup.*.sub.1](x; n)}.sub.n[greater than or equal to]p], deg [[OMEGA].sup.*.sub.1](x; n) = 1, n [greater than or equal to] p, such that

[[OMEGA].sup.*.sub.1](x; n)[B.sub.n](x) = [Q.sub.n+1](x) + [[theta].sup.*.sub.n,n][Q.sub.n](x), n [greater than or equal to] p. (4.6)

We write

[[theta].sup.*.sub.n,n] = [[gamma].sub.n]/[[lambda].sub.n,n-1], n [greater than or equal to] p, [[OMEGA].sup.*.sub.1](x; n) = x + [v.sup.*.sub.n,0, (4.7)

where

[v.sup.*.sub.n,0] = [[gamma].sub.n]/[[lambda].sub.n,n01] + [[lambda].sub.n+1,n] - [[beta].sub.n], n [greater than or equal to] p. (4.8)

Example. In order to illustrate the result of Proposition 4.1, we study the structure relation characterizing a semi-classical polynomial sequence, [{[B.sub.n]}.sub.n[greater than or equal to] 0, orthogonal with respect to the linear functional u solution of the functional equation

u' + [??]u = 0, (4.9)

where [??](x) = -[ix.sub.2] + 2x - i([alpha] - 1) and with regularity condition [alpha] [not member of] [[union.sub.n][greater than or equal to] 0] [E.sub.n], where [E.sub.0] = {[alpha] [member of] C : F([alpha]) = 0}, F([alpha])[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and for each integer n [greater than or equal to] 1, [E.sub.n] = {[alpha] [member of] C : [XI].sub.n]([alpha]) =0}. Here, [XI].sub.n]([alpha]) is the Hankel determinant associated with u. Notice that u is a semi-classical linear functional of class one [10].

The recurrence coefficients [[beta].sub.n] and [[gamma].sub.n+1], n [greater than or equal to] 0, of the sequence [{[B.sub.n]}.sub.n[greater than or equal to] 0] are determined by the system [10] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.10)

The sequence [{[B.sub.n]}.sub.n[greater than or equal to]0] is characterized by the following structure relation [10] :

[B.sup.[l].sub.n](x) = [B.sub.n](x) - i[gamma]n[gamma[n+1]/n + 1] [B.sub.n-1] (x), n [greater than or equal to] 1. (4.11)

Thus, taking into account [[lambda].sub.n,n-1] = i[gamma]n[gamma]n+1/n+1] [not equal to] 0, n [greater than or equal to] 1, we deduce a strictly finite-type relation between the sequences [{[B.sub.n]}.sub.n[greater than or equal to] 0] and [{[B.sup.[1].sub.n]}.sub.n[greater than or equal to] 0] with index s = 1, with respect to [phi[(x) = 1,

From Proposition 4.1. we get the following inverse relation, for n [greater than or equal to] 1,

(x + [v.sup.*.sub.n,0])[B.sub.n](x) = [B.sup.[1].sub.n+1](x) + i(n+1)/[gamma]n+1 [B.sup.[1].sub.n](x), (4.12)

where [v.sup.*.sub.n,0] = i(n+1)/[gamma]n+1 - i[gamma]n+1[gamma]n+2/n + 2 - [[beta].sub.n], n [greater than or equal to] 1. The sequence {B~}n>0 could be characterized by a relation as (4.12). It is the aim of the following result.

Proposition 4.2. Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be a MOPS satisfying (1.5). Then the following statements are equivalent.

i) There exists a set of non-zero complex numbers [{[[lambda].sub.n,n-1]}.sub.np[greater than or equal to]l] such that, for n [greater than or equal to] 1,

[B.sup.[1].sub.n](x) = [B.sub.n](x) + [[lambda].sub.n,n-1] [B.sub.n-1] (x). (4.13)

ii) There exists a set of complex numbers [{[??].sub.n]}.sub.n[greater than or equal to]0], with [[??].sub.n] [not equal to] 0, n [greater than or equal to] 1, and [[??].sub.0] = 0, such that for n [greater than or equal to] 0,

(x + [gamma]n+1/[??].sub.n+1]+[[??].sub.n]-[[beta].sub.n])[B.sub.n](x) = [B.sup.[1].sub.n+1](x)+[[??].sub.n][B.sup.[1].sub.n](x). (4.14)

Proof. Assume that i) holds. From Proposition 4.1, we get

(x + [gamma]n+1/[??]n+1] + [[??].sub.n]-[[beta].sub.n])[B.sub.n](x) = [[B.sui.[1].suib.n+1](x)[[??].sub.n][B.sup.[1].sub.n](x), n [greater than or equal to] 1,

where [[??].sub.n] = [[gamma].sub.n][[lambda].sup.-1.sub.n,n-1], n [greater than or equal to] 1. For n = 1, in (4.13) we obtain [[lambda].sub.1,0] = [[beta].sub.0]-[[beta].sub.1]/2. Then, [[gamma]1/[??]1 = [[beta].sub.0] - [[beta].sub.1]/2. Hence, (x+ [gamma]1/[??]1 - [[beta].sub.0](x) = x - [[beta].sub.0] + [[beta].sub.1]/2 = [B.sup.[1].sub.1](x) + [[??].sub.0][B.sup.[1].sub.0] (x), i.e. [[??].sub.0] = 0. Thus, ii) holds. Conversely, let us take [[lambda].sub.n,n-1] = [gamma]n/[??]n, n [greater than or equal to] 1, and consider the MPS [{[A.sub.n]}.sub.n[greater than or equal to] 0 defined by

[A.sub.n](x) = [B.sub.n](x) + [[lambda].sub.n,n-1][B.sub.n-l](x), n [greater than or equal to] 1. (4.15)

From Proposition 4.1, we get

(x + [v.sup.*.sub.n,0])[B.sub.n])[B.sub.n](x) = [A.sub.n+1](x) + [[theta].sup.*.sub.n,n][A.sub.n](x), n [greater than or equal to] 1,

where [v.sup.*.sub.n,0] = [gamma]n+1/[??]m+1 + [??]n - [[beta].sub.n], n [greater than or equal to] 1, and [[theta].sup.*.sub.n,n] = [gamma]n/[[lambda].sub.n,n-1] = [[??].sub.n], n [greater than or equal to] 1. From the assumption ii) and the previous relation, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3) (5.4)

The determinants associated with (5.1)-(5.2) are

[[DELTA].sub.0](0,2) = [[DELTA].sub.1](0,2) = 0, [[DELTA].sub.n](0, 2) = [[lambda].sub.n,n-2]([[lambda].sub.n+l,n-1] - [[gamma].sub.n]n), n [greater than or equal to] 2. (5.5)

As a consequence of Theorem 3.2, where t = 0 and s = 2, we have the following result

[A.sub.n+1](x) + [[??].sub.n], [A.sub.n](x) = [B.sup.[1].sub.n+1](x) + [[??].sub.n][B.sup.[1].sub.n](x), n [greater than or equal to] 1.

Equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But, from (4.15) for n = 1 we get [Ai.sub.l](x) = x - [[beta].sub.0] + [gamma]1/[[??].sub.1].

From (4.14), with n = 0, we get [B.sup.[1].sub.1](x) = x - [[beta].sub.0] + [gamma]1/[??]1].

Hence, [A.sub.n](x) = [B.sup.[1].sub.n](x), n [greater than or equal to] 0. Thus according to (4.15), i) holds.

5. The case (t, s) = (0, 2)

Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be a MOPS with respect to the linear functional u and satisfying (1.5). Consider the following finite-type relation between [{[B.sub.n]}.sub.n[greater than or equal to] 0] and [{[Q.sub.n]}.sub.n[greater than or equal to] 0], with index s = 2, with respect to [phi](x) = 1, for n [greater than or equal to] 0,

[Q.sub.n](x) = [B.sub.n](x)+ [[lambda].sub.n,n-1] [B.sub.n-1](x) + [[lambda].sub.n,n-2](x), (5.1)

[there exists]r[greater than or equal to] 2, [[lambda].sub.r,r-2] [not equal to] 0. (5.2)

From Lemma 2.1, for every system of complex numbers [([[theta].sub.n,v]).sup.n+2.sub.v=n], n [greater than or equal to] 0, where [[theta].sub.n,n+2] = 1, n [greater than or equal to] 0 and [[theta].sub.r,r] [not equal to] 0, there exists a unique MPS [{[[OMEGA].sub.2](x; n)}.sub.n[greater than or equal to] 0], where [[OMEGA].sub.2](x; n) = [chi square] + [v.sub.n,l]x + [v.sub.n,0], n [greater than or equal to] 0, and a unique system of complex numbers, [([[zeta].sup.[0].sub.n,v]).sup.n-1.sub.v=n-2], n [greater than or equal to] 0, such that

Proposition 5.1. Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be a MOPS and [{[Q.sub.n]}.sub.n[greater than or equal to] 0] be the MPS satisfying (5.1)-(5.2). For every fixed integer p [greater than or equal to] 2, the following statements ate equivalent

i) [[lambda].sub.n,n-2]([[lambda].sub.n+l,n-1] - [[gamma].sub.n]) [not equal to] 0, n [greater than or equal to] p.

ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example. Let [{[B.sub.n]}.sub.n[greater than or equal to] 0] be the sequence of monic polynomials, orthogonal with respect to the linear functional u such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

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 [[beta].sub.n] = 0, n [greater than or equal to] 0, and where [[gamma].sub.n+1], n [greater than or equal to] 0, are given by a non-linear recurrence relation (see [3] and [15])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The sequenee [{[B.sub.n]}.sub.n[greater than or equal to] 0] satisfies the following structure relation ( see [3])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.8)

From (5.3), with [Q.sub.n](x) = [B.sup.[1].sub.n](x), n [greater than or equal to] 0, and the fact that the polynomial sequences [{[B.sub.n]}.sub.n[greater than or equal to] 0] and [{[B.sup.[1].sub.n]}.sub.n[greater than or equal to] 0] are symmetric, i.e, [B.sub.n](-x) = [(-1).sup.n][B.sub.n](x), n [greater than or equal to] 0, we get, for n [greater than or equal to] 0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, the determinants associated with (5.8) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Acknowledgements: The first author (FM) was supported by Direccion General de Investigacion (Ministerio de Educacion y Ciencia) of Spain under grant MTM 2006-13000-C03-02. The work of second author (RS) was supported by Entreprise Kilani at Gabes and Institut Superieur de Gestion de Gabes, Tunisie.

Recibido el 16 de diciembre de 2007

Aceptado para su publicacion el 14 de marzo de 2008

References

[1] M. Alfaro, F. Marcellan, A. Pena & M. L. Rezola, On linearly related orthogonal polynomials and their functionals, J. Math. Anal. Appl. 287 (2003), 307-319.

[2] M. Alfaro, F. Marcellan, A. Pena & M. L. Rezola, On rational transformations of linear functionals. A direct problem, J. Math. Anal. Appl. 298 (2004), 171-183.

[3] A. Cachafeiro, F. Marcellan & J.J. Moreno-Balcazar, On asymptotic properties of Freud-Sobolev orthogonal polynomials, J. Approx. Theory 125 (2003), 26-41.

[4] T. S. Chihara, "An Introduction to Orthogonal Polynomials", Gordon and Breach, New York, 1978.

[5] A. M. Delgado & F. Marcellan. Companion linear functionals and Sobolev inner products: a case study, Meth. Appl. Anal. 11 (2004), 237-266.

[6] F. Marcellin, M. Alfaro & M. L. Rezola, Orthogonal polynomials on Sobolev spaces: Old and new directions, J. Comput. Appl. Math. 48 (1993), 113-131.

[7] F. Marcellan & J. C. Petronilho, Orthogonal polynomials and coherent pairs: The classical case, Indag. Math. (NS), 6 (1995), 287-307.

[8] F. Marcellin, T. E. Perez & M. A. Pinar, Orthogonal polynomials on weighted Sobolev spaces: the semi-classical case, Ann. Numer. Math. 2 (1995), 93-122.

[9] F. Marcellan & J. C. Medem, Q-Classical orthogonal polynomials: a very classical approch, Elect. Trans. on Nuroer. Anal. 9 (1999), 112-127.

[10] P. Maroni, Un exemple d'une suite orthogonal semiclassique de classe un. In Publ. Labo. d'Analyse Numerique, Universite Pierre et Marie Curie, Paris. 89033 (1989).

[11] P. Maroni, Fonctions euleriennes. Polynomes orthogonaux classiques. In Techniques de l'ingenieur, A 154 (1994), 1-30.

[12] P. Maroni, Semi-classical character and finite-type relations between polynomial sequences. J. Appl. Num. Math. 31 (1999), 295-330.

[13] P. Maroni &: R. Sfaxi, Diagonal orthogonal polynomial sequences, Meth. Appl. Anal. 7 (2000), 769-792.

[14] H. G. Meijer, Determination of all Coherent Pairs. J. Approx. Theory, 89 (1997), 321-343.

[15] P. Nevai, Orthogonal Polynomials associated with exp(--[x.sup.4]). Proc. Canad. Math. Soc. 3 (1983). 263-265.

Francisco Marcellan (1) & Ridha Sfaxi (2)

(1) Departamento de Matematicas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain. Correo electronico: pacomarc@ing.uc3m.es

(2) Departement des Methodes Quantitatives, Institut Superieur de Gestion de Gabes, Avenue Jilani Habib 6002, Gabes, Tunisie. Correo electronico: ridhasfaxi@yahoo.fr 2000 Mathematics Subject Classification: 42C05, 33C45.

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Title Annotation: | Matematicas |
---|---|

Author: | Marcellan, Francisco; Sfaxi, Ridha |

Publication: | Revista de la Academia Colombiana de Ciencias Exactas, Fisicas y Naturales |

Date: | Jun 1, 2008 |

Words: | 5837 |

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