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Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle.

1 Introduction

Let F be a Caratheodory function in the Laguerre-Hahn class, i.e., satisfying a Riccati differential equation with polynomial coefficients (see [4])

zAF' = [BF.sup.2] + CF + D, A [not equivalent to] 0. (1)

A first approach to the analysis of Caratheodory functions satisfying this type of differential equations and to the analysis of its corresponding sequences of orthogonal polynomials was done by Alfaro and Marcellan in [2]. We remark that the Laguerre-Hahn class on the unit circle includes some well known classes, such as the Laguerre-Hahn affine class on the unit circle (which corresponds to the case B [equivalent to] 0 in (1)), the semi-classical class on the unit circle (which corresponds to the case B [equivalent to] 0 and D a specific polynomial in (1)), and the class of second degree functionals on the unit circle. It also includes linear fractional transformations of Laguerre-Hahn Caratheodory functions (see [3,4, 7, 8]).

The motivation for the study here presented comes from several applications related with orthogonal polynomials on the unit circle and also on the real line. In what concerns to the orthogonality on the real line we note the works of Magnus [18], Maroni [21, 22] and Hahn [15, 16]. In [18] Magnus used the theory of Laguerre-Hahn orthogonal polynomials (the "Riccati model") in the study of the convergence of Jacobi continued fractions. This was done, first, by considering a modified approximant which satisfies a Riccati differential equation and, then, by estimating the error behavior with the help of appropriate linear differential equations which are satisfied by a sequence of Laguerre-Hahn orthogonal polynomials (see [15,16]). See also the example in [18, section 5], showing the use of the Riccati model in disordered systems analysis. In [21, 22], Maroni studies the Laguerre-Hahn class on the real line from an algebraic point of view, putting emphasis on the distributional equations for the corresponding forms defined in the linear space of real polynomials; some modifications that preserve the Laguerre-Hahn character are studied (in [4] the analogue of these results are established for Laguerre-Hahn functionals on the unit circle).

Let us now return to the orthogonality on the unit circle. Since the Laguerre-Hahn class on the unit circle contains linear fractional transformations of Caratheodory functions which satisfy Riccati type differential equations, then it is a suitable class to study some transformations concerned with the measure of orthogonality or with the orthogonal polynomials when one starts, for example, with Laguerre-Hahn affine orthogonal polynomials, or with orthogonal polynomials associated with second degree Caratheodory functions. Here are some examples:

a) shift perturbation of the reflection coefficients of the orthogonal polynomials (see [23]);

b) backward extension or modification of a finite number of places of the reflection coefficients of the orthogonal polynomials (see [23]);

c) rational perturbation of the measure of orthogonality (see [5, 6]).

In this paper we aim to obtain a characterization of the Laguerre-Hahn Caratheodory functions and a representation for the corresponding sequences of orthogonal polynomials on the unit circle. We will see that, also on the unit circle, the first order differential relations satisfied by Laguerre-Hahn orthogonal polynomials play an important role. In fact, a key result of our paper is the equivalence between (1) and the following matrix Sylvester differential equations for

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[phi].sub.n], {[[OMEGA].sub.n]}, and {[Q.sub.n]} are the corresponding sequences of orthogonal polynomials, of polynomials of the second kind, and functions of the second kind, respectively, [B.sub.n] and C are matrices of order two with polynomial elements, and I is the identity matrix of order two (see Theorem 3).

As a consequence of the referred equivalence, we obtain a characterization of polynomials which are orthogonal with respect to a semi-classical weight, in terms of first order linear systems of differential equations (see Theorem 4). These systems are similar to the ones derived in [10, 17] (see also [19]). But here it is well to emphasize that, in those papers, the authors went further and studied the dynamics of the linear systems of differential equations subject to deformations of the semi-classical weight, thus showing the occurrence of Schlesinger systems as well as Painleve equations.

The equivalence between (1) and (2) allows us to give a representation for {[Y.sub.n]} in terms of the solutions of two linear differential systems, zACJ = CL and [zAP'.sub.n] = [B.sub.n][P.sub.n], as [Y.sub.n] = [P.sub.n][L.sup.-1], [for all]n [greater than or equal to] 1 (see Theorem 5). Furthermore, the characterization for semi-classical polynomials previously obtained will help us to establish that the Caratheodory function F in (1) is a linear fractional transformation of a semi-classical Caratheodory function, say F (see Theorem 6), and we give a representation for {[Y.sub.n]} in terms of the semi-classical orthogonal polynomials corresponding to F (see Theorem 7).

This paper is organized as follows. In section 2 we give the definitions and state the basic results which will be used in the forthcoming sections. In section 3 we establish the equivalence between (1) and the matrix Sylvester differential equations (2). In section 4 we establish a characterization of semi-classical orthogonal polynomials on the unit circle in terms of first order linear differential systems. In section 5 we solve the matrix Sylvester differential equations from section 3, [zAY'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C, with the help of the the results previously obtained for semi-classical orthogonal polynomials. Thus, we determine a representation for the solution, [Y.sub.n], in terms of sequences of semi-classical orthogonal polynomials on the unit circle. Finally, in section 6, an example is presented.

2 Preliminary results

Let [mu] be a probability measure with infinite support on the unit circle T = {[e.sup.i[theta]] : [theta] [member of] [0,27[phi]]}. The corresponding sequence of orthogonal polynomials, called orthogonal polynomials on the unit circle (with respect to [mu]), is defined by

1/2[pi] [[integral].sup.2[pi].sub.0] [[phi].sub.n] ([e.sup.i[theta]])[[bar.[phi]].sub.m]{[e.sup.- i[theta]])d[mu]{[theta]) = [h.sub.n][[delta].sub.n,m][h.sub.n] [not equal to] 0, n, m [member of] N.

If [mu] is absolutely continuous with respect to d[theta], associated with a weight w, i.e., d[mu]([theta]) = w(theta]) d theta], then we say that {[[phi].sub.n]} is orthogonal with respect to w. If each [[hpi].sub.n] is monic, then {[[phi].sub.n]} will be called a monic orthogonal polynomial sequence, and it will be denoted by MOPS.

Given a measure [mu], the function F defined by

F(z) = 1/2[pi] [[integral].sup.2[pi].sub.0] [e.sup.i[theta]] + z/[e.sup.i[theta]] - z d [mu] ([theta] (3)

is a Caratheodory function, i.e., it is an analytic function in [??] = {z [member of] [??] : [absolute value of z] < 1} such that F(0) = 1 and Re(F) > 0 for [absolute value of z] < 1. The converse result also holds, since any Caratheodory function has a representation (3) for a unique probability measure [mu] on T (see, for example, [25]). In addition, it is well known that d[[mu].sub.r] ([theta]) = ReF([re.sup.i[theta]]) d[theta] converge weakly to d[mu] when r [up arrow] 1, [lim.sub.r[arrow]1 ReF([re.sup.i[theta]) = ReF([e.sup.i[theta]) exists a.e. for [theta] [member of] [0,2[pi]], and if d[mu]([theta]) = w([theta])d[theta] + d[[mu].sub.s]([theta]), with d[[mu].sub.s] the singular part, then

w([theta]) = ReF([e.sup.i[theta]).

Given a sequence of monic polynomials {[[phi].sub.n]} orthogonal with respect to u, the associated polynomials of the second kind are given by

[OMEGA].sub.0](z) = 1, [OMEGA].sub.n] (z) = 1/2[pi][[integral].sup.2[pi].sub.0] [e.sup.i[theta]] + z/e.sup.i[theta]] + z ([[phi].sub.n]{[e.sup.i[theta]]) - [[phi].sub.n]{z)) d[mu]([theta]), [for all]n [member of] N,

and the functions of the second kind are given by

We define the following matrices which will be used throughout the paper,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [[phi].sup.*.sub.n] and [[OMEGA].sup.*.sub.n] denote the reciprocal polynomial of [[phi].sub.n] and [[OMEGA].sub.n], respectively, and [Q.sup.*.sub.n] (z) = [[OMEGA].sup*.sub.n]On (z) - F(z)[[phi].sup.*.sub.n] (z). We recall that the reciprocal polynomial [p.sup.*] of a polynomial p of exact degree n is defined by [p.sup.*](z) = [z.sup.n][bar.p]{1/z).

The sequences {[[phi].sub.n]}, {[[OMEGA].sub.n]} and {[Q.sub.n]} satisfy recurrence relations and coupled relations which we use in the matrix form, as given in the following theorem (see [14]).

Theorem 1 (cf. [12,13, 24]). Let F be a Caratheodory function and {[[phi].sub.n]}, {[[OMEGA].sub.n]}, and {[Q.sub.n]} the corresponding MOPS on the unit circle, the sequence of associated polynomials of the second kind, and the sequence of the functions of the second kind, respectively. Let {[Y.sub.n]} and {[Q.sub.n]} be the sequences defined in (4). Then, the following relations hold, [for all]n [member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, [for all]n [member of] N,

[[phi].sup.*.sub.n](z)[[OMEGA].sub.n](z) + [[phi].sub.n](z)[[OMEGA].sup.*.sub.n](z) = [2h.sub.n][z.sup.n], (7)

[[phi].sup.*.sub.n](z)[[OMEGA].sub.n](z) + [[phi].sub.n](z)[[OMEGA].sup.*.sub.n](z) = [2h.sub.n][z.sup.n], (8)

with [h.sub.n] = [[PI].sup.n.sub.k]n=1](1 - [[absolute value of [a.sub.k]].sup.2]).

Let [H.sub.0](z) = [[summation].sup.+[infinity].sub.j=0] [b.sub.j][z.sup.j], [absolute value of z] < 1, [H.sub.[infinity]] = [[summation].sup.+[infinity].sub.j=0] [b.sub.j][z.sup.-j], [absolute value of z] > 1. We will write [H.sub.0](z) = O[z.sup.k] or [H.sub.[infinity]] (z) = O[z.sup.-k] if [b.sup.0] = ... = [b.sup.k-1] = 0, k [member of] N.

Corollary 1. Let {[[phi].sub.n]} be a MOPS on the unit circle and {[[OMEGA].sub.n]} be the corresponding sequence of functions of the second kind. Then, [for all]n [member of] N,

[Q.sub.n] (z) = [2h.sub.n][z.sup.n] + O ([z.sup.n+1]), [absolute value of z] < 1,

[Q.sub.n] (z) = [2a.sub.n+1][h.sub.n][z.sup.-1] + O ([z.sup.-2]), [absolute value of z] > 1,

[Q.sup.*.sub.n] (z) = [2[[bar.a].sub.n+1][h.sub.n][z.sup.n+1] + O ([z.sup.n+2]), [absolute value of z] > 1

[Q.sup.*.sub.n] (z) = [2[h.sub.n] + O ([z.sup.-1]), [absolute value of z] < 1

with [a.sub.n+1] = [[phi].sub.n+1](0), [h.sub.n] = [[PI].sup.n.sub.k=1](1 - [absolute value of [[a.sub.k].sup.2]).

Corollary 2. Let {[[phi].sub.n]} be a MOPS on the unit circle and {[[OMEGA].sub.n]} be the corresponding sequence of associated polynomials ofthe second kind. Then, the following holds:

a) If there exists k [member of] N such that [[phi].sub.k]([alpha]) = [[OMEGA].sub.k]([alpha]) = 0, then [alpha] = 0;

b) If there exists k [member of] N such that [[phi].sub.k]([alpha]) = [[OMEGA].sub.k]([alpha]) = 0, then [alpha] = 0.

Theorem 2 (Geronimus, [11]). Given a sequence of complex numbers ([a.sub.n]) satisfying [absolute value of [a.sub.n]] < 1, [for all]n [member of] N, let {[[phi].sub.n]} and {[[OMEGA].sub.n]} be the sequences of polynomials defined by the recurrence relation (5), and let F be the corresponding Caratheodory function. Then, the sequence defined for n [greater than or equal to] 1 by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

converges uniformly to F(z), on compact subsets of D.

3 Characterization in terms of matrix Sylvester differential equations

Hereafter, I denotes the identity matrix of order two.

Theorem 3. Let F be a Caratheodory function and {[Y.sub.n]} and {[Q.sub.n]} the corresponding sequences defined by (4). The following statements are equivalent:

a) F satisfies the differential equation with polynomial coefficients

zAF' = [BF.sup.2] + CF + D; (9)

b) {[Y.sub.n]} and {[Q.sub.n]} satisfy the Sylvester differential equations

[zAY'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C (10)

[zAQ'.sub.n] = ([B.sub.n] + (BF + C/2) I) [Q.sub.n], n [member of] N, (11)

where [B.sub.n] are matrices of bounded degree polynomials,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Proof. a) [??] b).

Let F satisfy (9). Firstly we obtain (10). This will be done by dividing the proof in two parts: in the first part we deduce the equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

and in the second part we deduce the equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [l.sub.n,1], [l.sub.n,2], [THETA].sub.n,1], [THETA].sub.n,2] are polynomials whose degrees are bounded by a number independent of n. These two systems of equations can be written in the matrix form (1o), with [B.sub.n] and C given by (12) and (13), respectively.

First part. If we substitute F = [Q.sub.n]/[[phi].sub.n] - [Q.sub.n]/[[phi].sub.n] (cf. (6)) in zAF' = [BF.sup.2] + CF + D we obtain

zA ([Q.sub.n]/[[phi]].sub.n] - [Q.sub.n]/[[phi]].sub.n])' = B [([Q.sub.n]/[[phi]].sub.n] - [Q.sub.n]/[[phi]].sub.n]).sup.2] + C ([Q.sub.n]/[[phi].sub.n] - [Q.sub.n]/[[phi].sub.n]) + D,

i.e.,

zA ([Q.sub.n]/[[phi].sub.n])' - B[Q.sub.n]/[[phi].sub.n] ([Q.sub.n]/[[phi].sub.n] - 2[Q.sub.n]/[[phi].sub.n]) - C[Q.sub.n]/[[phi].sub.n]

= zA([Q.sub.n]/[[phi].sub.n])' + B[([Q.sub.n]/[[phi].sub.n]).sup.2] - C([Q.sub.n]/[[phi].sub.n]) + D,

Therefore we have

{zA([Q.sub.n]/[[phi].sub.n])' + B[([Q.sub.n]/[[phi].sub.n]).sup.2] - C([Q.sub.n]/[[phi].sub.n]) + D} [[phi].sup.2.sub.n] = [[THETA].sub.n] (16)

with

[[THETA].sub.n] = {zA([Q.sub.n]/[[phi].sub.n])' + B[Q.sub.n]/[[phi].sub.n] ([Q.sub.n]/[[phi].sub.n] - 2[Q.sub.n]/[[phi].sub.n]) - C([Q.sub.n]/[[phi].sub.n])}[[phi].sup.2.sub.n].

From (16) it follows that [[??].sub.n] is a polynomial. From the asymptotic expansion of [Q.sub.n] in [absolute value of z] < 1 (see Corollary 1), and since the left side of (16) is a polynomial, we get

[[THETA].sub.n](z)= [z.sub.n] [[??].sup.1.sub.n](z),

with [[??].sup.1.sub.n] a polynomial. From the asymptotic expansion of [Q.sub.n] in [absolute value of z] > 1 (see Corollary 1) it follows that [[??].sup.1.sub.n] has bounded degree,

deg([[??].sup.1.sub.n]) = max{deg(zA) - 2,deg(B) - 1,deg(C) - 1}, [for all]n [member of] N.

Thus, (16) becomes

{zA([Q.sub.n]/[[phi].sub.n])' + B[([Q.sub.n]/[[phi].sub.n]).sup.2] - C([Q.sub.n]/[[phi].sub.n]) + D} [[phi].sup.2.sub.n] = [z.sup.n][[??].sup.1.sub.n]

Using (7) in the previous equation we obtain

{zA([Q.sub.n]/[[phi].sub.n])' + B[([Q.sub.n]/[[phi].sub.n]).sup.2] - C([Q.sub.n]/[[phi].sub.n]) + D} [[phi].sup.2.sub.n] = [[THETA].sub.n,1] ([[phi].sub.n][[OMEGA].sup.*.sub.n] + [[OMEGA].sub.n][[phi].sup.*.sub.n]

where [[THETA].sub.n,1] = [[??].sup.1.sub.n]/([2h.sub.n]).

Consequently, [for all]n [member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We distinguish the following cases (see Corollary 2):

a) [[phi].sub.n] and [OMEGA].sub.n] have no common roots, [for all]n [member of] N, i.e., [[phi].sub.n](0) [not equal to] 0, [for all]n [member of] N;

b) there exists a finite number of indexes k [member of] N such that [[phi].sub.k] and [[OMEGA].sub.k] have common roots, i.e., [[phi].sub.k](0) = [[OMEGA].sub.k](0) = 0 for a finite number of k's;

c) there exists [n.sub.0] > 1 such that [[phi].sub.n](0) = 0, [for all]n [greater than or equal to] [n.sub.0].

Case a) If [phi].sub.n] and [[OMEGA].sub.n] have no common roots, [for all]n [member of] N, then we conclude that there exists a polynomial [l.sub.n,1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

and we obtain (14). Moreover, [l.sub.n,1] has bounded degree,

deg([l.sub.n,1]) = max{deg(zA) - 1, deg(B), deg(C), deg(D),}, [for all]n [member of] N.

Case b) We first assume that [[phi].sub.1](0) [not equal to] 0, ..., [[phi].sub.k-1](0) [not equal to] 0, and k is the first index such that [[phi].sub.k](0) = 0. Thus, [[phi].sub.n] and [[OMEGA].sub.n] have no common roots for n = 1, ..., k - 1. From case a), equations (17) hold for n = 1, ..., k - 1. Let us write (17) to k - 1 and multiply the resulting equation by z, to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By substituting

[[OMEGA].sub.k](z) = z[[OMEGA].sub.k-1](z), [[OMEGA].sup.*.sub.k](z) = [[OMEGA].sup.*.sub.k-1](z), z[[OMEGA]'.sub.k-1] (z) = [[OMEGA]'.sub.k](z) - [[OMEGA].sub.k-1](z)

and

[[phi].sub.k](z) = k[[phi].sub.k-1](z), [[phi].sup.*.sub.k](z) = [[phi].sup.*.sub.k-1](z), z[[phi]'.sub.k-1](z) = [[phi]'.sub.k](z) - [[phi].sub.k-1](z)

in previous equations, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we obtain (14) to n = k with [l.sub.k,1] = [l.sub.k-1,1] + A and [[THETA].sub.k,1] = z[[THETA].sub.k-1,1].

Furthermore, if [[phi].sub.k+1](0) = ... = [[phi].sub.k+k0](0) = 0, [[phi].sub.k+k0+1](0) [not equal to] 0 to some [k.sub.0] [member of] N, then, using the same method as before, the differential relations (14) are obtained for n = k + 1, ..., k + [k.sub.0], with

[l.sub.n,1] = [l.sub.k-1,1] + (n - k + 1) A, [[THETA].sub.n,1] = [z.sup.n-k+1][[THETA].sub.k-1,1], n = k + 1, ..., k + [k.sub.0].

Case c) If [[phi].sub.n](0) = 0, [for all]n [greater than or equal to] [n.sub.0], then [[phi].sub.n] and [[OMEGA].sub.n] are polynomials of the Bernstein-Szego type,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Applying the same method as before, we conclude that equations (14) hold, [for all]n [member of] N, and, for n [greater than or equal to] [n.sub.0], [l.sub.n,1] and [[THETA].sub.n,1] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Second part. If we substitute F = [[OMEGA].sup.*.sub.n/[[phi].sup.*.sub.n] - [Q.sup.*.sub.n]/[[phi].sup.*.sub.n] (cf. (6)) in zAF' = [BF.sup.2] + CF + D and proceed as in the first part, we obtain (15) with

deg([l.sub.n,2]) = max{deg(zA) - 1, deg(B), deg(C), deg(D)}, [for all]n [member of] N, deg([[THETA].sub.n,2]) = max{deg(zA) - 1, deg(B), deg(C)}, [for all]n [member of] N.

Let us now obtain (11). Taking derivatives on [Q.sub.n] = [[OMEGA].sub.n] + [[phi].sub.n]F and [Q.sup.*.sub.n] = [[OMEGA].sup.*.sub.n] - [[phi].sup.*.sub.n]F (cf. (6)) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we use (9), (14) and (15) in the previous equations, then (11) follows. b) [??] a).

Taking into account (6), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [for all]n [member of] N, the equation (11) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From (10) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking into account that [Y.sub.n] is nonsingular, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since C is given by (13), zAF' = [BF.sup.2] + CF + D follows.

The following formula for tr([B.sub.n]) was given in [20] for a particular case of a semi-classical sequence of orthogonal polynomials on the unit circle.

Corollary 3. Under the conditions of the previous theorem, the matrices [B.sub.n] given by (12) satisfy

zA[A'.sub.n] = [B.sub.n][A.sub.n] - [A.sub.n][B.sub.n-1], n [greater than or equal to] 2, (18)

tr([B.sub.n]) = nA, n [member of] N, (19)

det([B.sub.n]) = det([B.sub.1]) - A [n-1.summation over (k=1)] [l.sub.k,2,] n [greater than or equal to] 2, (20)

where tr([B.sub.n]) and det([B.sub.n]) denote the trace and the determinant of [B.sub.n], respectively, and

det{[B.sub.1]) = A(2zA[[bar.a].sub.1] - [h.sub.1](D + B) + C([absolute value of [a.sub.1]].sup.2] + 1)/(2[h.sub.1]) + BD - [C.sup.2]/4, (21)

[a.sub.1] = [[phi].sub.1](0), [h.sub.1] = 1 - [absolute value of [a.sub.1]].sup.2].

Proof. To obtain (18) we take derivatives on [Y.sub.n] = [A.sub.n][Y.sub.n-1] and substitute [Y'.sub.n] = [A'.sub.n][Y.sub.n-1] + [A.sub.n][Y'.sub.n-1] in (10), zA[Y'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C. Therefore, we get

zA[A'.sub.n][Y.sub.n-1] + zA[A.sub.n][Y'.sub.n-1] = [B.sub.n][Y.sub.n] - [Y.sub.n]C.

Using (10) for n - 1 in the previous equation we get

zA[A'.sub.n][Y.sub.n-1] + [A.sub.n]([B.sub.n-1][Y.sub.n-1] - [Y.sub.n-1]C) = [B.sub.n][Y.sub.n] - [Y.sub.n]C.

Using the recurrence relation (5) we obtain

zA[A'.sub.n][Y.sub.n-1] + [A.sub.n]([B.sub.n-1][Y.sub.n-1] - [Y.sub.n-1]C) = [B.sub.n][A.sub.n][Y.sub.n-1] - [A.sub.n][Y.sub.n-1]C,

i.e.,

zA[A'.sub.n][Y.sub.n-1] = ([B.sub.n][A.sub.n] - [A.sub.n][B.sub.n-1]) [Y.sub.n-1].

Since [Y.sub.n] is nonsingular, for all n [member of] N and z [not equal to] 0, we obtain (18).

To deduce (19) we use equations (14) and (15),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we multiply the previous equations by [[OMEGA].sup.*.sub.n], [[phi].sup.*.sub.n], [[phi].sub.n] and [[OMEGA].sub.n], respectively, we obtain, after summing,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

i.e.,

zA([[phi].sub.n][[OMEGA].sup.*.sub.n] + [[phi].sup.*.sub.n][[OMEGA].sub.n])' = ([l.sub.n,1] + [l.sub.n,2]) ([[phi].sub.n][[OMEGA].sup.*.sub.n] + [[phi].sup.*.sub.n][[OMEGA].sub.n]).

Thus,

zA([[phi].sub.n][[OMEGA].sup.*.sub.n] + [[phi].sup.*.sub.n][[OMEGA].sub.n])' = tr([B.sub.n])([[phi].sub.n][[OMEGA].sup.*.sub.n] + [[phi].sup.*.sub.n][[OMEGA].sub.n]).

If we use (7) in the previous equation then we get (19).

We now establish (2o). From (18) we obtain, for n [greater than or equal to] 2,

det([B.sub.n][A.sub.n]) = det(zA[A'.sub.n] + [A.sub.n][B.sub.n-1]).

Taking into account that [B.sub.n] is given by (12) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

det([B.sub.n]) det([A.sub.n]) = z(1 - [[absolute value of [a.sub.n]].sup.2]) (det([B.sub.n-1]) + A[l.sub.n-1,2]), [for all] n [greater than or equal to] 2.

Since det ([A.sub.n]) = z(1 - [[absolute value of [a.sub.n]].sup.2], then the last equation is equivalent, if z [not equal to] 0, to

det([B.sub.n]) = det([B.sub.n-1]) + A[l.sub.n-1,2], [for all]n [greater than or equal to] 2.

Consequently, we obtain (20). Moreover, if we compute det([B.sub.1]) by taking n = 1 in (10), we obtain (21).

Remark. (18) is equivalent to the following equations, for all n [member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

4 A characterization of semi-classical orthogonal polynomials on the unit circle

In this section we derive a characterization for sequences of semi-classical orthogonal polynomials on the unit circle.

Definition 1 (cf. [26]). Let [mu] be a measure supported on the unit circle given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[lambda].sub.k] [greater than or equal to] 0, [absolute value of [z.sub.k]] = 1, k = 1, ..., N, N [member of] N. The weight w (or the measure [mu]) is semi-classical if there exist polynomials A, C such that

dw([theta])/d[theta]/w([theta] = C(z)/A(z), z = [e.sup.i[theta]]. (23)

The corresponding sequence of orthogonal polynomials is called semi-classical.

For our purposes, we will consider the analytic continuation of the weight w to an annulus {z : [[epsilon].sub.1] < [absolute value of z] < [[epsilon].sub.2]} and, in order to simplify the notation, we still denote this analytic continuation by w = w(z). Thus, the equation (23) is now equivalent to

w'(z)/w(z) = -1C(z)/zA(z) (': = d/dz). (24)

It is well known that the corresponding Caratheodory function satisfies a first order linear differential equation

zA(z)F'(z) = -iC(z)F(z) + D(z),

where D is a polynomial (see [26]). Moreover, the converse holds for a specific polynomial D depending on A, C (see [3, 7]).

We will need the lemma that follows (see [9]).

Lemma 1. Let X and M be matrix functions of order two such that X' = MX. Then,

(det(X))' = tr(M) det(X). (25)

The theorem that follows is a generalization of a result for semi-classical orthogonal polynomials on the real line established in [19]. Moreover, it shows that the necessary condition given in [3, Theorem 5] for a MOPS on the unit circle to be semi-classical is also sufficient.

Theorem 4. Let {[[phi].sub.n]} be a MOPS with respect to a weight w, {[Q.sub.n]} be the sequence of functions of the second kind, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

if, and only if, [[??].sub.n] satisfies

zA[[??]'.sub.n] = ([B.sub.n] - C/2 I)[[??].sub.n], [for all]n [member of] N, (27)

where [B.sub.n] is the matrix associated with the equation zAF' = CF + D satisfied by the corresponding Caratheodory function.

Proof. Let w satisfy w'/w = C/(zA) and let the corresponding F satisfy zAF' = CF + D.

From Theorem 3 the following two equations hold,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)

where [B.sub.n] are the matrices associated with zAF' = CF + D. Moreover, as w'/w = C/(zA), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

If we substitute (28) in (30) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

Finally, from (29) and (31), the differential system (27) follows.

We now prove the converse.

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies (27) then, from Lemma 1, we obtain

(det([[??].sub.n]))' = tr([B.sub.n] - C/2I/zA)det([[??].sub.n]).

From (8) we get det([[??].sub.n]) = 2[h.sub.n][z.sup.n]/w, thus the last equation is equivalent to

w'/w = nA - tr([B.sub.n] - C/2I)/zA.

If we use tr([B.sub.n]) = nA (cf. (19)) in the previous equation then we get w'/w = C/zA, and we conclude that w is given by (26).

5 Solutions of the Sylvester differential equations

In this section we solve the Sylvester differential equations (1o), zA[Y'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C, [for all]n [member of] N. The result that comes next is a particular case of a result on matrix Riccati equations known as Radon's Lemma (see [1]).

Theorem 5. Let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D and [B.sub.n], C be the corresponding matrices given by (12) and (13), respectively. Let G [subset] C be a domain not containing the zeros of zA, and [z.sub.0] [member of] G. If L (L nonsingular) and [P.sub.n] satisfy, [for all]n [member of] N,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

then the corresponding sequence {[Y.sub.n]} associated with F, defined in (4), has the following representation in G,

[Y.sub.n](z) = [P.sub.n](z) [L.sup.-1](z), [for all]n [member of] N. (34)

Proof. To zAF' = [BF.sup.2] + CF + D we associate (10), zA[Y'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C, with [B.sub.n] and C given by (12) and (13), respectively (see Theorem 3). Let L and [P.sub.n] satisfy (32) and (33), respectively. Let us see that that [Y.sub.n] = [P.sub.n][L.sup.-1] is the solution of zA[Y'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C. Taking into account that

zA([P.sub.n][L.sup.-1])' = zA[P'.sub.n][L.sup.-1] + zA[P.sub.n]([L.sup.-1])'

and ([L.sup.-1])' = -[L.sup.-1]L'[L.sup.-1], from (33) we get

zA([P.sub.n][L.sup.-1])' = [B.sub.n][P.sub.n][L.sup.-1] - zA[P.sub.n][L.sup.-1]L'[L.sup.-1].

Using (32) in the previous equation we get

zA([P.sub.n][L.sup.-1])' = [B.sub.n][P.sub.n][L.sup.-1] - [P.sub.n][L.sup.-1]CL[L.sup.-1],

i.e., [Y.sub.n] = [P.sub.n][L.sup.-1] satisfies zA[Y'.sub.n] = [B.sub.n][Y.sub.n] - [Y.sub.n]C, and the assertion follows.

Remark. The solution of (32) is given by L(z) = L(z)[L.sup.0], with L a fundamental matrix of the differential system (32) satisfying zAL' = CL, and [L.sup.0] = L[([z.sub.0]).sup.-1]. The solution of (33) is given by [P.sub.n](z) = [P.sub.n](z)[P.sup.0.sub.n], with [P.sub.n] a fundamental matrix of (33) satisfying zA[P'.sub.n] = [B.sub.n][P.sub.n], and [P.sup.0.sub.n] satisfying [P.sub.n]([z.sub.0])[P.sup.0.sub.n] = [Y.sub.n]([z.sub.0]), i.e., [P.sup.0.sub.n] = [([P.sub.n]([z.sub.0])).sup.-1] [Y.sub.n]([z.sub.0]). Thus, if we substitute L and [P.sub.n], given as above, in (34), the solution of the Sylvester differential equations (10) becomes

[Y.sub.n](z) = [P.sub.n](z)[E.sub.n][L.sup.-1](z) (35)

with

[E.sub.n] = [([P.sub.n]([z.sub.0])).sup.-1] [Y.sub.n]([z.sub.0])L([z.sub.0]). (36)

5.1 Solution of (32)

We search for a matrix L of order 2 satisfying zA(z)L'(z) = C(z)L(z), with C given in (13).

Lemma 2. Let L be a fundamental matrix of solutions of (32). Then, det(L(z)) = det(L([z.sub.0])).

Proof. From Lemma 1 (cf. (25)) we have

(det(L))' = tr(C)/zA det(L).

Since tr(C) = , it follows that(det(L))' = 0, i.e.,

det(L) = c, c [member of] C.

Thus, det(L(z)) = det(L([z.sub.0])), for some [z.sub.0] [member of] C.

Lemma 3. Let C be the matrix defined by (13). Then,

(a) [C.sup.2] = [beta]I, [beta] = [(C/2).sup.2] - BD;

(b) The eigenvalues of C are [+ or -] [square root of [beta]];

(c) [V.sub.[square root of [beta]] = span{[[D C/2 - [square root of [beta]].sup.T]} is the eigenspace corresponding to [square root of [beta]] and

[V.sub.-[square root of [beta]] = span{[[D C/2 + [square root of [beta]].sup.T]} is the eigenspace corresponding to - [square root of [beta]].

In what follows, [L.sub.1], [L.sub.2] are column vectors of size 2.

Lemma 4. Let L = [[L.sub.1] [L.sub.2]] be a fundamental matrix of (32). Then,

zA[L'.sub.1] = [square root of [beta]][L.sub.1] + zA[c.sub.1][V.sub.-[square root of [beta]], (37)

zA[L'.sub.2] = [square root of [beta]][L.sub.2] + zA[c.sub.2][V.sub.[square root of [beta]], (38)

with [c.sub.1], [c.sub.2] functions.

Proof. From (32) it follows that

{C + [square root of [beta]]I)([L'.sub.1] - [square root of [beta]]/zA [L.sub.1]) = [0.sub.2x1], (39)

{C - [square root of [beta]]I)([L'.sub.2] - [square root of [beta]]/zA [L.sub.2]) = [0.sub.2x1], (40)

Since the eigenvalues of C are [+ or -] [square root of [beta]] and the corresponding eigenvectors are [V.sub.[square root of [beta]]] and [V.sub.[square root of -[beta]]], from (39) and (40) we obtain, respectively,

([L'.sub.1] - [square root of [beta]]/zA [L.sub.1]) = [c.sub.1](z)[V.sub.-[square root of [beta]]]

([L'.sub.2] - [square root of [beta]]/zA [L.sub.2]) = [c.sub.2](z)[V.sub.[square root of [beta]]]

where [c.sub.1], [c.sub.2] are functions. Thus, (37) and (38) follow.

5.2 Solution of (33)

We search for matrices [P.sub.n] of order two satisfying, for each n [member of] N,

zA[P'.sub.n] = [B.sub.n][P.sub.n]. (41)

Hereafter we will consider [z.sub.1] [member of] C and [??] a polynomial such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined (in suitable domains).

Lemma 5. Let [B.sub.n] be the matrices given in (12), let A, [??] be polynomials. [[??].sub.n] is a solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (42)

if, and only if, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a solution of (41).

Proof. Let [[??].sub.n] be a solution of (42). Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [[??].sub.n] satisfies (42), then we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies (41). Analogously one proves the converse.

Taking into account the previous lemma, we will solve (41) searching for a solution {[P.sub.n]} given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], n [member of] N, where [[??].sub.n] satisfies (42). Furthermore, we will search for [[??].sub.n] given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [for all]n [member of] N, where {[[??].sub.n]} is a MOPS on the unit circle with respect to a weight function [??], and {[[??].sub.n]} is the corresponding sequence of functions of the second kind.

Let us remark that, using the same arguments as the ones used in the proof of Theorem 4, from zA[[??]'.sub.n] = ([B.sub.n] - [??]/2I)[[??].sub.n] we get

[??]'/[??] = nA- tr([B.sub.n] - [??]/2 I)/zA,

and since tr([B.sub.n]) = nA, there follows [??]'/[??] = [??]/zA, thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (43)

Henceforth,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (44)

with [??] given by (43).

Remark. According to Theorem 4, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[??].sub.n] is associated with the equation for the corresponding Caratheodory function, say [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], thus depending on A, [??], [??]. On the other hand, [B.sub.n] of (42) depend on A, B, C, D. As it will be seen in Lemma 7, this is possible because the polynomials B, C, D depend on [??], [??].

Lemma 6. Let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D and {[[phi].sub.n]} the corresponding MOPS. For all n [member of] N, let [P.sub.n] be a fundamental matrix of the corresponding differential system (33). If [P.sub.n] is given by (44), where {[[??].sub.n]} is the MOPS with respect to the weight w, then the following equations hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (45)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (46)

Proof. (45) is a consequence of the recurrence relations for {[[??].sub.n]} (see Theorem 1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

We now establish (46). Since [P.sub.n] satisfies zA[P'.sub.n] = [B.sub.n][P.sub.n], then by substituting [P.sub.n] = [[??].sub.n][P.sub.n-1] in the previous equation, there follows

zA[A'.sub.n][P.sub.n-1] + [[??].sub.n]zA[P'.sub.n-1] = [B.sub.n][[??].sub.n][P.sub.n-1], n [greater than or equal to] 2.

Using zA[P'.sub.n-1] = [B.sub.n-1][P.sub.n-1] in the last equation we get

zA[[??].sub.n][P.sub.n-1] + [[??].sub.n][B.sub.n-1][P.sub.n-1] = [B.sub.n][[??].sub.n][P.sub.n-1].

Thus,

(zA[[??]'].sub.n] + [A.sub.n][B.sub.n-1])[P.sub.n-1] = [B.sub.n][[??].sub.n][P.sub.n-1].

Since [P.sub.n] is nonsingular (det([P.sub.n]) [not equal to] 0, [for all]n [member of] N, [for all]z [not equal to] 0) then

zA[[??]'.sub.n] + [[??].sub.n][B.sub.n-1] = [B.sub.n][[??].sub.n]

follows, and we obtain (46).

Remark. From (18) and (46) we get the equations

zA([A.sub.n] - [[??].sub.n])' = [B.sub.n]([A.sub.n] - [[??].sub.n]) - ([A.sub.n] - [[??].sub.n])[B.sub.n-1], n [greater than or equal to] 2.

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

Hereafter we will denote linear fractional transformations T(F) = a + bF/c + dF by [T.sub.(a,b;c,d)](F).

Theorem 6. Let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D, and {[[phi].sub.n]} be the corresponding MOPS. Let [P.sub.n], n [member of] N, be a fundamental matrix of the differential system (33) given by (44), and [??] be the corresponding Caratheodory function. Then, there exists a unique linear fractional transformation, [T.sub.(a,b;c,d)], with a, b, c, d [member of] P and ad - bc [not equivalent to] 0, such that F = [T.sub.(a,b;c,d)]([??]).

Proof. To prove that F is a linear fractional transformation of [??], we begin by establishing that the reflection coefficients of {[[phi].sub.n]} and {[[??].sub.n]}, i.e., [a.sub.n] = [[phi].sub.n](0) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], differ only in a finite number of indexes.

Let us write [[lambda].sub.n] = [a.sub.n] - [[??].sub.n], [for all]n [member of] N. First we establish that Z = {n [member of] N : [[lambda].sub.n] [not equal to] 0} is a finite set. In fact, if Z was not finite, for example, Z [equivalent to] N, then [[lambda].sub.n] = 0, [for all]n [member of] N. But from (47) we would obtain

[l.sub.n,1] = [l.sub.n-1,2], [for all]n [member of] N.

Substituting in (22), we would obtain

[[THETA].sub.n,1] = z[[THETA].sub.n-1,1] [for all]n [member of] N,

hence

[[THETA].sub.n,1] = [z.sup.n][[THETA].sub.1,1] [for all]n [member of] N.

But this is a contradiction to the fact that deg([[THETA].sub.n]) is bounded. Therefore, Z [??] N. On the other hand, if we consider, without loss of generality, the case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then we will obtain the same conclusion.

To conclude that F is a rational transformation of [??] of the referred type, we take into account its representation in continued fraction given in Theorem 2. To establish the uniqueness of [T.sub.(a,b;c,d)] we remind that the inverse of [T.sub.(a,b;c,d)], ad - bc [not equal to] 0, is given by [T.sub.(a, -c; -b,d)]. Therefore, if [T.sub.1] and [T.sub.2] are two linear fractional transformations such that [T.sub.1]([??]) = [T.sub.2]([??]), then the composition [T.sup.-1.sub.2] o [T.sub.1] satisfies ([T.sup.-1.sub.2] o [T.sub.1])([??]) = [??], and thus we obtain [T.sup.-1.sub.2] o [T.sub.1] = id, i.e., [T.sub.1] = [T.sub.2]. Hence, the uniqueness of T is established.

5.3 Determination of the polynomial [??]

In what follows we determine the polynomial [??] which defines {[P.sub.n]} given in (44).

Lemma 7. Under the conditions of the previous theorem, let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D, let [??] be a polynomial which defines a weight [??] given by (43), and let [??] be the Caratheodory function associated with [??]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let us consider the first order linear differential equation for [??],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

Then, the following relations hold:

B = ([[alpha].sub.2][[beta]'.sub.2] - [[alpha]'.sub.2][[beta].sub.2])zA + [[alpha].sub.2][[beta].sub.2][??] + [[beta].sup.2.sub.2][??], (49)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (50)

D = ([[alpha].sub.1][[beta]'.sub.1] - [[alpha]'.sub.1][[beta].sub.1])zA + [[alpha].sub.1][[beta].sub.1][??] + [[beta].sup.2.sub.1][??], (51)

where we have considered, without lost of generality, [[alpha].sub.2][[beta].sub.1] - [[alpha].sub.1]][[beta].sub.2] = 1.

Proof. Since w'/[??] = [??]/(zA) (cf. (43)), then [??] is semi-classical. Therefore, the corresponding [??] satisfies (48), with [??] a polynomial (see [3, 7]).

Let us write F = [[alpha].sub.1] - [[beta].sub.1][??]/-[[alpha].sub.2] = [[beta].sub.2][??], i.e., [??] [[alpha].sub.1] + [[alpha].sub.2]F/[[beta].sub.1] + [[beta].sub.2]F. Using [??] = [[alpha].sub.1] + [[alpha].sub.2]F/[[beta].sub.1] + [[beta].sub.2]F in (48), it follows that

zA([[alpha].sub.2][[beta].sub.1] - [[alpha].sub.1][[beta].sub.2])F' = [B.sub.2][F.sup.2] + [C.sub.2]F + [D.sub.2], (52)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, F satisfies zAF' = [BF.sup.2] + CF + D and (52), thus it follows that

ZA([[alpha].sub.2][[beta].sub.1] - [[alpha].sub.1][[beta].sub.2])/zA = [B.sub.2]/B [C.sub.2]/C = [D.sub.2]/D.

Therefore, if ([[alpha].sub.2][[beta].sub.1] - [[alpha].sub.1][[beta].sub.2]) = 1, then

B = [B.sub.2], C = [C.sub.2], D = [D.sub.2],

and (49)-(51) follow.

According to Theorem 6, for each polynomial [??] defining a weight [??] by (43) and {[P.sub.n]} as in (44), there exists a unique linear fractional transformation T such that F = T([??]), with F the Caratheodory function associated with [??]. In this issue, we pose the question: being [[??].sub.1] and [[??].sub.2] polynomials (defining weights of the same type as in (43)) and [[??].sub.1], [[??].sub.2] the corresponding Caratheodory functions such that F is a linear fractional transformation of [[??].sub.i], i = 1, 2, to obtain relations between [[??].sub.1] and [[??].sub.2]. The next lemma gives us an answer.

Lemma 8. Under the same conditions of the previous lemma, let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D. Let [[??].sub.1], [[??].sub.2] be polynomials defining semi-classical weights of the type (43), and let [F.sub.1] and [F.sub.2] be the corresponding Caratheodory functions, non rational, satisfying

zA[F'.sub.1] = [[??].sub.1][F.sub.1] + [[??].sub.1], (53)

zA[F'.sub.2] = [[??].sub.2][F.sub.2] + [[??].sub.2]. (54)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the transformations such that [T.sub.1]([F.sub.1]) = F, [T.sub.2]([F.sub.2]) = F. If we assume, without loss of generality, that [[alpha].sub.2][[beta].sub.1] - [[alpha].sub.1][[beta].sub.2] = 1, [[gamma].sub.2][[eta].sub.1] - [[gamma].sub.1][[eta].sub.2] = 1, then the following relations take place:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (55)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

Proof. Since F = [T.sub.1]([F.sub.1]) with [F.sub.1] satisfying (53), from previous lemma we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Also, since F = [T.sub.2]([F.sub.2]) with [F.sub.2] satisfying (54), from previous lemma we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, (55)-(57) follow.

We now state the main result of this section, a representation formulae for {[Y.sub.n]}, defined in (4), associated with a Caratheodory function F that satisfies zAF' = [BF.sup.2] + CF + D.

Theorem 7. Let F be a Caratheodory function satisfying zAF' = [BF.sup.2] + CF + D, A, B, C, D [member of] P, and let {[Y.sub.n]} be the corresponding sequence given by (4). Then, there exists a polynomial [??] (defined by Lemmas 7 and 8), and a weight [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where {[[??].sub.n]} is the MOPS with respect to [??], {[[??].sub.n]} is the sequence of functions of the second kind associated with {[[??].sub.n]}, En are the matrices defined in (36), and L is a fundamental matrix of(32).

Proof. These equations are a direct application of Theorem 5, namely (35).

6 Example

Let us consider the sequence of Jacobi orthogonal polynomials on the unit circle, {[[phi].sub.n]}, with parameters [alpha] = [beta], [??] the corresponding Caratheodory function. Let {[[OMEGA].sub.n]} be the sequence of associated polynomials of the second kind and F be the corresponding Caratheodory function. F satisfies (see [4])

z([z.sup.2] - 1)F' = -2 [alpha][c.sub.0]([z.sup.2] - 1)[F.sup.2] - 2[alpha]([z.sup.2] + 1)F,

where [c.sub.0] is the moment of order zero of the Jacobi measure on the unit circle.

Taking into account Theorem 5, firstly we will solve the following differential systems:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

z([z.sup.2] - 1)[P'.sub.n] = [B.sub.n][P.sub.n]. (59)

In what follows we consider a complex domain G such that {0, 1, -1} [subset not equal to] G, and a [z.sub.0] in G.

Lemma 9. The fundamental matrix of solutions of (58) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we obtain a solution of (59). Takin into account Theorem 4, henceforth we will consider [??] as polynomial and we will solve (59) searching for a solution [[??].sub.n] given by (44), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the MOPS with respect to [??], {[[??].sub.n]} the corresponding sequence of functions of the second kind, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, F is a linear fractional transformation of [??] given by F = 1/[??] (see, for example, [23, 25]), with [??] satisfying (see [26])

z([z.sup.2] - 1)[??]' = 2[alpha]([z.sup.2] + 1)[??] + 2[alpha][c.sub.0]([z.sup.2] - 1).

Therefore, by Lemma 7, [??] = 2[alpha]([z.sup.2] + 1) follows, and consequently we obtain [??] = [(([z.sup.2] - 1)/z).sup.2[alpha]].

From Theorem 7, the following representation for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

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* This work was supported by CMUC, Department of Mathematics, University of Coimbra. The second author was supported by FCT, Fundacao para a Ciencia e Tecnologia, with grant ref. SFRH/BD/25426/2005.

The authors would like to thank Professor Jank, by the insight into the theory of matrix Riccati equations.

Received by the editors November 2008.

Communicated by A. Bultheel.

2000 Mathematics Subject Classification : 33C45, 39B42.

A. Branquinho: CMUC and Department of Mathematics, Coimbra University, Largo D. Dinis, 3001-454 Coimbra, Portugal.

email:ajplb@mat.uc.pt

M.N. Rebocho: CMUC, University of Coimbra, Largo D. Dinis, 3001-454 Coimbra, Portugal,

and

Department of Mathematics, University of Beira Interior, 6201-001 Covilha, Portugal.

email:mneves@mat.ubi.pt
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Author:Branquinho, A.; Rebocho, M.N.
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