z z’x P+ P+ P + · · ·

V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275“295 277

where

« «

0 0 1 0 0

5

¬ · ¬ ·

ex t d (x)

P =¬0 0 ·; Cn = ¬ 0 ·

1 0

d (x; t) = ;

ex5 t d 1; 1 (x)

0 z 0 0 ’bn+2

and d is a matrix (p — q) = (3 — 2) of positive measures.

What is a matrix continued fraction is explained in Section 5 [6,13,15]. So Theorem B means that

from the initial conditions bn+3 (0), we compute the continued fraction (i.e., solve the direct spectral

problem), then write the result as the Cauchy transform of the matrix measure. Then using simple

dynamics of this measure, we decompose the Cauchy integral into a continued fraction (solve the

inverse spectral problem) and get bn+3 (t) for t ∈ [0; +∞[.

Theorem B has two aspects, the ÿrst is algebraic, namely the algorithm of construction of the

matrix continued fraction, the second is analytic, i.e., we get a positive measure. This aspect deals

with zeros properties of the convergents of the continued fraction (see Section 6), which are Hermite“

PadÃ approximants [10,12], of the function F on the left-hand side of (1). So, we begin our paper

e

by the deÿnition of these approximants in Sections 2 and 3.

Theorems A and B are in fact proved simultaneously. In Theorem B we have the solution of

system (E) on [0; +∞[ but not the uniqueness. A local theorem gives existence and uniqueness on

some segment [0; t0 ], and its length depends only on the constant M . If bn+3 (t0 ) are known to be

positive bounded by the same constant M , then bn+3 (t) can be extended on the segment [t0 ; 2t0 ] and

so on. Such information is obtained from Theorem B, using techniques of genetic sums (Section 4)

[14]. So in complement of the preceding results, we also have

0 ¡ bn+3 (t)6M; t ∈ [0; +∞[; n¿0:

To get the dynamics of the spectral measure d in (1), we need the following di erential equation

satisÿed by its power moments, Sn = x5n d (x; t) = (Sn j ), i = 1; 2; 3; j = 1; 2 the matrices being

i;

Sn = Sn+1 ’ Sn S1 1 ;

1;

n = 0; 1; : : : : (2)

How to prove that systems (E) and (2) are equivalent ? We use the standard method of Lax pairs.

We consider the bi-inÿnite matrix

«

0 0 0 a0 ···

¬ ·

¬ ·

..

¬ ·

.

··· ··· ···

¬ ·

A=¬ ·; (3)

¬ bp ·

0 ap ···

¬ ·

.. ..

. .

where for system (E) an = 1 (but it is possible to consider another normalization), and look for an

inÿnite matrix B such that system (E) can be rewritten in the form

A = [A; B];

where [A; B] = AB ’ BA. There exists such matrix B for our system and it is constructed in Section 7.

Then we have

(An ) = [An ; B]

278 V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275“295

which is proved by recurrence. Then the resolvent operator can be expanded in a neighbourhood of

inÿnity,

∞

An

’1

Rz = (zI ’ A) = ;

z n+1

n=0

hence

Rz = [Rz ; B]: (4)

But the function F on the left-hand side of (1) is the top left corner of Rz (as fi; j = (Rz ei’1 ; ej’1 ))

and (4), in particular, means, with deÿned in (7)

S0

’ S1 1 F;

1;

F = z5 F ’

z +1

which is equivalent to (2).

We see that the solution of system (E) is found through the theory of Hermite“PadÃ approximants

e

for which new results are proved: genetic sums, zeros properties.

2. The matrix Hermite“PadÃ problem

e

We deÿne a matrix of power series with complex coe cients

«

f1; 1 f1; q

···

∞

¬ · fn j

i;

F=¬ ·; i; j

··· f (z) = :

z n+1

n=0

fp; 1 fp; q

···

In matrix form

∞

fn

fn = (fn j )i=1; ::: ; p;

i;

F= ; j=1; ::: ; q :

z n+1

n=0

We now consider the Hermite“PadÃ problem (H“P) for F: for any n¿0, two regular multiindices

e

are deÿned, i.e., N = (n1 ; : : : ; np ) and M = (m1 ; : : : ; mq ) such that

p

ni = n; n1 ¿n2 ¿ · · · ¿np ¿n1 ’ 1;

1

q

mi = n + 1; m1 ¿m2 ¿ · · · ¿mq ¿m1 ’ 1

1

(the Hermite“PadÃ problem can be considered for any indices, but we will restrict the study to the

e

case of regular multiindices). We look for polynomials Hn1 ; : : : ; Hnq not equal zero simultaneously, of

1 p

degree not greater than m1 ’ 1; : : : ; mq ’ 1, such that for some polynomials Kn ; : : : ; Kn the following

V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275“295 279

conditions hold:

R1 = f1; 1 Hn1 + · · · + f1; q Hnq ’ Kn = O(1=z n1 +1 );

1

n

.

.

.

Rp = fp; 1 Hn1 + · · · + fp; q Hnq ’ Kn = O(1=z np +1 ):

p

n

There always exists a nontrivial solution to this problem.

Considering the vectors Rn = (R1 ; : : : ; Rp )t , Kn = (Kn ; : : : ; Kn )t and Hn = (Hn1 ; : : : ; Hnq )t , we get with

1 p

n n

matrix notation

Rn = FHn ’ Kn = O(1=z N +1 );

deg Hn 6M ’ 1:

We consider C[X ]q the sets of column vectors of size q and their canonical basis

« « « «

1 0 0 x

¬· ¬· ¬· ¬·

¬0· ¬1· ¬0· ¬0·

¬· ¬· ¬· ¬·

h0 = ¬ . · ; h1 = ¬ . · ; hq’1 = ¬ . · ; hq = ¬ . · · · · ;

···;

¬· ¬· ¬· ¬·

¬.· ¬.· ¬.· ¬.·

. . . .

0 0 1 0

i.e., if n = rq + s, s ¡ q, the components of hn are zero except the component of index s which is

equal to xr , the components being numbered from 0 to q ’ 1. The same thing is done for C[X ]p

and its canonical basis is denoted by en , n¿0, column vectors of size p.

The matrix moments fn deÿne the generalized Hankel H matrix in block form and in scalar

form, and for any n positive Hn is the n — n minor in the upper left-hand corner

« «

h0; 0 h0; 1 h0; 2

f0 f1 f2 ··· ···

¬ · ¬ ·

¬ · · · · = ¬ h1; 0 ····

h1; 1 h1; 2

H = ¬ f1 f2 f3 (5)

·¬ ·

. . . . . .

. . . . . .

. . . . . .

and we deÿne the bilinear functional

: C[X ]p — C[X ]q ’ C; = hl; k ;

(el ; hk ) = el ; hk

where l; k¿0.

The H“P problem is equivalent to the orthogonality relations

Hn ∈ Span (h0 ; : : : ; hn ); (ek ; Hn ) = 0; k = 0; : : : ; n ’ 1: (6)

We will restrict the study to the “(p|q) symmetric” case, i.e., the functions fi; j are, up to a shift,

functions of z p+q . Let a system S of p — q real sequences (Sn j )n be given for i = 1; : : : ; p; j = 1; : : : ; q.

i;

The parameters p and q are supposed to be relatively prime, so there exist integers u and v such

280 V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275“295

that

up ’ vq = 1:

According to what is found for the Weyl function F, we deÿne the constants

i; j

= (j ’ i)(u + v)mod (p + q); i = 1; : : : ; p; j = 1; : : : ; q (7)

and the fi; j are

∞

Sn j

i;

i; j

f (z) = :

z (p+q)n+ i; j +1

n=0

∞

Example 2.1. In the scalar case p = q = 1, f1; 1 is a symmetric function f1; 1 (z) = Sn 1 =z 2n+1 .

1;

n=0

Example 2.2. In the vector case p = 2, q = 1,

∞ ∞

Sn 1

1;

Sn 1

2;

1; 1 2; 1

f (z) = ; f (z) = :

z 3n+1 z 3n+2

n=0 n=0

Example 2.3. In the matrix case p = 3, q = 2, we get

∞ ∞

Sn 1

1;

Sn 2

1;

1; 1 1; 2

f (z) = ; f (z) = ;

z 5n+1 z 5n+3

n=0 n=0

∞ ∞

Sn 1

2;

Sn 2

2;

2; 1 2; 2

f (z) = ; f (z) = ;

z 5n+4 z 5n+1

n=0 n=0

∞ ∞

Sn 1

3;

Sn 2

3;

3; 1 3; 2

f (z) = ; f (z) = :

z 5n+2 z 5n+4

n=0 n=0