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As deÃ¿ned in (5), Hn are the main n Ã— n minors of H, and we get the classical deÃ¿nition.

DeÃ¿nition 2.4. System S is called nonsingular or positive if, respectively,

Hn = 0; Hn Â¿ 0:

Lemma 2.5. If S is nonsingular then

â€¢ The (H â€“P) problem has a unique solution Hn (up to normalization).

â€¢ deg Hn = n with respect to the basis hk .

â€¢ Hn is Zp+q -invariant; i.e.; Hn âˆˆ Span{hk ; k = n mod(p + q)}.

The proof is a consequence of Cramerâ€™s rule.

V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275â€“295 281

3. Generalized Jacobi matrix

We suppose the system S is nonsingular, and we get [12]

h0; 0 h0; 1 h0; n

Â·Â·Â·

h1; 0 h1; 1 h1; n

Â·Â·Â·

un . . .

. . .

Hn = ; (8)

. . .

Hn

hnâˆ’1; 0 hnâˆ’1; 1 hnâˆ’1; n

Â·Â·Â·

h0 h1 Â·Â·Â· hn

where the last row of the determinant is composed of vectors. The nonzero constants un are the

leading coe cients of Hn (and may be changed in order to normalize the vector polynomials in one

way or the other)

Hn = un hn + Â· Â· Â· :

In [12] we have already got the following.

Theorem 3.1. There exists a unique set of complex coe cients a(m) ; m = âˆ’p; : : : ; q; nÂ¿0; n + mÂ¿0

n

such that the sequence of vector polynomials (Hk )k is the unique solution of the recurrence relation

an Hn+q + Â· Â· Â· + a(1) Hn+1 + a(0) Hn + an Hnâˆ’1 + Â· Â· Â· + a(âˆ’p) Hnâˆ’p = xHn

(q) (âˆ’1)

(9)

n n n

with the initial conditions

Hâˆ’p = Â· Â· Â· = Hâˆ’1 = 0;

h0; 0 h0; j

Â·Â·Â·

Â·Â·Â·

uj

Hj = ; j = 0; : : : ; q âˆ’ 1:

Hj hjâˆ’1; 0 hjâˆ’1; j

Â·Â·Â·

h0 Â·Â·Â· hj

In particular;

un

(q)

an = ; nÂ¿0;

un+q

un+p Hn+p+1 Hn

a(âˆ’p) = ; nÂ¿0: (10)

n+p

un Hn+p Hn+1

Because here, only the symmetric case is considered, the result is simpliÃ¿ed.

Theorem 3.2. The sequence Hn is the unique solution of the recurrence relation

xHn = an Hn+q + bn Hnâˆ’p ; (11)

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

with initial conditions

Hâˆ’p = Â· Â· Â· = Hâˆ’1 = 0; H j = uj h j ; j = 0; : : : ; q âˆ’ 1:

an and bn being deÃ¿ned as the an ; a(âˆ’p) of the previous theorem.

(q)

n

The proof consists of the fact that H = xHn âˆ’ bHnâˆ’p satisÃ¿es the same orthogonality relations as

Hn+q if enâˆ’p ; H = 0 (which deÃ¿nes b). It is also a consequence of the fact that xHn ; Hn+q ; Hnâˆ’p

are the only polynomials of (9) that depend on hk ; k = n mod(p + q).

The recurrence relation (9) can be written in matrix form as

AH = xH; (12)

where H is the inÃ¿nite column vector (H0 ; H1 ; : : :)t (each term being a vector, H could be written

as a scalar matrix (âˆž Ã— q)) and A a scalar inÃ¿nite band matrix with two nonzero diagonals,

ï£« ï£¶

0 Â·Â·Â· Â·Â·Â· a0 0 0

ï£¬ ï£·

ï£¬. ï£·

ï£¬. 0ï£·

ï£¬. 0 Â·Â·Â· Â·Â·Â· a1 ï£·

ï£¬ ï£·

ï£¬. .. ï£·

..

ï£¬. .ï£·

.

ï£¬. ï£·

A=ï£¬ ï£·:

ï£¬ ï£·

ï£¬ bp Â·Â·Â· Â·Â·Â· ï£·

ï£¬ ï£·

ï£¬ ï£·

ï£¬0 ï£·

bp+1 Â·Â·Â· Â·Â·Â·

ï£¬ ï£·

ï£ ï£¸

..

.

0 0

From the relations in the previous theorem, an and bn are positive if and only if the system S is

positive (the vector polynomials Hn being considered with a positive normalization constant un ). If,

for the polynomials Hn , we take the monic polynomials (un = 1), then matrix A satisÃ¿es an = 1.

Another normalization will be used. If the leading coe cients of Hn are un , deÃ¿ned by

p=(p+q)

Hn

un =

Hn+1

then [13]

(an Â· Â· Â· an+pâˆ’1 )q = (bn+p Â· Â· Â· bn+p+qâˆ’1 )p : (13)

Such a matrix (two diagonals satisfying (13), an Â¿ 0; bn Â¿ 0) generalizes the symmetric case and

is called a generalized Jacobi matrix. If J is the set of such matrices, and S the set of the

positive systems S normalized by S0 1 = 1, we have constructed a one-to-one correspondence (by

1;

the generalized Shohatâ€“Favard theorem [12]) from S to J. In this case we will use the following

representation of the parameters an and bn , Hn0 denoting the monic polynomials:

mn = en ; Hn0 ; un = (1=mn )p=(p+q) ;

p=(p+q)

mn+1 mn+q

= (cn Â· Â· Â· cn+qâˆ’1 )p=(p+q) = ( n )p=(p+q) ;

cn = ; an =

mn mn

q=(p+q)

mn+p

= (cn Â· Â· Â· cn+pâˆ’1 )q=(p+q) = (Ã¿n+p )q=(p+q) :

bn+p =

mn

V. Sorokin, J. Van Iseghem / Journal of Computational and Applied Mathematics 122 (2000) 275â€“295 283

The parameters cn are uniquely deÃ¿ned if p and q are relatively prime.

4. Genetic sums

The aim of this section and the following is to give a representation by a matrix continued

fraction, generalizing the S-fraction of Stieltjes, of the resolvent function of the operator deÃ¿ned in

the standard basis of the Hilbert space l2 (0; âˆž) by the bi-inÃ¿nite, (p + q + 1)-band matrix A with

p + q âˆ’ 1 zero intermediate diagonals. The detailed proofs can be found in [14]. The matrix A

is taken with the normalization giving an = 1, i.e., the matrix is deÃ¿ned by Ai; i+q = 1, Ai+p; i = bi ;

equivalently the operator is deÃ¿ned by

A(ei ) = bi+p ei+p ; i Â¡ q; A(ei ) = eiâˆ’q + bi+p ei+p ; iÂ¿q;

where the constants bi are a sequence of nonzero complex numbers. We assume that

sup(|bi |) Â¡ + âˆž

to deal with bounded operators, and that p and q are relatively prime (up âˆ’ vq = 1).

Let R(z) be the resolvent operator,

Rz = (zI âˆ’ A)âˆ’1 :

It is known [7] that the set of resolvent functions, the so-called Weyl functions

F = (fi; j ); fi; j (z) = Rz eiâˆ’1 ; ejâˆ’1 ; i = 1; : : : ; p; j = 1; : : : ; q

can be chosen as spectral data, su cient for the determination of the operator A given by the previous

(p + q + 1)-band matrix. The functions fi; j are analytic in the neighbourhood of inÃ¿nity, because

of the boundedness of A, and have power series expansions

âˆž

fn j

i;

i; j

f (z) = ; i = 1; : : : ; p; j = 1; : : : ; q:

z n+1

n=0

Hence, the formal solution of direct and inverse spectral problems means to Ã¿nd a constructive

procedure for the determination of the spectral data fi; j or fn j i = 1; : : : ; p; j = 1; : : : ; q and nÂ¿0

i;

from the operator data bn (nÂ¿p) and vice versa.

In the following, in order to give explicitly all the formulae, we will write everything for p = 3

and q = 2, but the results are general.

Because all the intermediate diagonals are zero, we are able to Ã¿nd a particular form for the

moments called genetic sums [15]. This form has already been found for the vector case, recovered

with q = 1 [1].

With the preceding notations, the moments, i.e., the coe cients fn j of each function can be

i;

computed in terms of the bi . Because the functions are found to be functions of the variable z p+q ,

for each function fi; j , all the coe cients, except a subsequence with indices n(i; j), are zero. The

results are as follows for p = 3, q = 2.

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

Theorem 4.1. The moments of the Weyl functions associated to the operator A are given by the

following: for all nÂ¿0,

1; 1

Sn 1 = fn(1; 1) = b3

1;

n(1; 1) = 5n; bi2 Â· Â· Â· bi2n ;

i2 i2n

1; 2

Sn 2 = fn(1; 2) = b3

1;

n(1; 2) = 2 + 5n; bi1 Â· Â· Â· bi2n ;

i1 i2n

2; 1

Sn 1 = fn(2; 1) = b4

2;

n(2; 1) = 3 + 5n; bi1 Â· Â· Â· bi2n ;

i1 i2n

2; 2

Sn 2 = fn(2; 2) = b4

2;

n(2; 2) = 5n; bi2 Â· Â· Â· bi2n ;

i2 i2n

3; 1

Sn 1 = fn(3; 1) =

3;

n(3; 1) = 1 + 5n; bi1 Â· Â· Â· bi3n ; i1 âˆˆ {3; 6};

i1 i2n

3; 2

Sn 2 = fn(3; 2) =

3;

n(3; 2) = 3 + 5n; bi b i1 Â· Â· Â· bi2n ; i âˆˆ {3; 6}

i i1 i2n

in all cases ik = ikâˆ’1 + p âˆ’ q; integer; varying such that 16ik 6ikâˆ’1 + p. In all other cases

fk j = 0.

i;

This means that fi; j are recovered as

âˆž

Sn j

i;

i; j

i; j p+qâˆ’ âˆ’1

f (z) = z :

(z (p+q) )(n+1)

n=0

The proof of these formulae can be found in [14]. The same method of proof leads to some

identities for the genetic sums.

To the sequence (bn )nÂ¿p is associated, for each pair (i; j); i = 1; : : : ; p; j = 1; : : : ; q the sequence

Sn j , nÂ¿0, and similarly Sn j (k), kÂ¿0, nÂ¿0 corresponding to the sequence (bn+k )nÂ¿p . Each family

i; i;

can be considered as the coe cients of a formal series and represented by this series, which deÃ¿nes

S i; j (k) and fi; j (k):

Sn j (k)

i;

i; j

S (k)(z) = ; i = 1; : : : ; p; j = 1; : : : ; q; kÂ¿0: (14)

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