Solvability of the Hankel determinant problem …

Solvability of the Hankel determinant problem for real sequences

Andrew Bakan  and  Christian Berg Institute of Mathematics
National Academy of Sciences of Ukraine
Tereschenkivska Street 3, Kyiv 01601, Ukraine
andrew@bakan.kiev.ua Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen
Denmark
berg@math.ku.dk
Abstract.

To each nonzero sequence of real numbers we associate the Hankel determinants of the Hankel matrices , , and the nonempty set . We also define the Hankel determinant polynomials , and , as the determinant of the Hankel matrix modified by replacing the last row by the monomials . Clearly is a polynomial of degree at most and of degree if and only if . Kronecker established in 1881 that if is finite then for each , where . By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence to be of the form , for a real sequence . This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial satisfying is preceded by a nonzero polynomial whose degree can be strictly less than and which has no common zeros with . As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that and for all .

Key words and phrases:
Hankel matrices, Frobenius rule, Kronecker theorem, orthogonal polynomials
1991 Mathematics Subject Classification:
Primary 44A60, 47B36; Secondary 15A15, 15A63

1. Introduction

We use the notation and . To a sequence of real numbers we associate the Hankel matrices , and the determinants , . In this way we get a mapping in the space of sequences of real numbers. We call this mapping the Hankel determinant transform. It was introduced and studied by Layman in [14] who emphasized that such a transform is far from being injective by proving that a sequence and its binomial transform defined by

have the same image under this mapping. Concerning the missing injectivity let us here just point out that the Hankel determinant transform of all the sequences , is .

Several authors have been concerned with the sign pattern of the sequence in order to use this for the determination of the rank and signature of the Hankel matrices. This is given in rules of e.g. Jacobi, Gundelfinger and Frobenius. See [9],[11] for a treatment of these questions, which become quite technical when zeros occur in the sequence .

The Hankel determinant problem for real sequences is to characterize the image in , i.e., to find a necessary and sufficient condition for a sequence to be of the form

(1.1)

with some sequence of real numbers. It turns out that such conditions are similar to those that were obtained by G.Frobenius [7, p.207] in 1894 for all possible signs of the numbers . His arguments were simplified by F.Gantmacher [9, p.348] in 1959 and by I.S. Iohvidov [11, (12.8), p.83] in 1982 in an essential way. The purpose of the present paper is to obtain a further simplification of the Frobenius reasoning by giving in Theorem 3 a new setting of the approach suggested by Iohvidov in [11, Chapter II]. This allows to give in Section 7 a self-contained proof of the following theorem.

Theorem 1.

Let be a sequence of real numbers and

If then the equation (1.1) is satisfied if and only if for all . If consists of distinct elements arranged in increasing order then the equation (1.1) is solvable if and only if the following Frobenius conditions (see [9, p.348]) hold

It follows from Theorem 1 that (1.1) is solvable if for all , and not solvable if . Furthermore, the condition for all is sufficient for the existence of at least one solution of (1.1).

Let us formulate an elementary result about existence and uniqueness of solutions to (1.1) and which is independent of Theorem 1. For this we need the following notation. For a determinant , we denote by , the determinant obtained by deleting the ’th row and ’th column of . For Hankel determinants we follow Frobenius [7, p.212] in writing , , i.e.,

(1.2)
Proposition 1.

Given two sequences of real numbers such that for all , there exists a unique sequence of real numbers such that

To see this we use the Laplace expansion of and along the last column and note that . This gives the following recurrence formulas

where , , depend only on , , is a function of and a function of . If are assumed to be given, these relations determine the sequence uniquely, and the assertion follows.

A complete description of all solutions of (1.1), when some of the numbers vanish, can be derived from the Frobenius results in [7], but this is of no relevance in the present context.

Let denote the set of all algebraic polynomials with real coefficients. Given a sequence of real numbers we introduce two sequences of polynomials in :

(1.3)

Note that , , and

(1.4)

The polynomials are called Hankel determinant polynomials with respect to the sequence . Let denote the linear functional determined by

(1.5)

Then

(1.6)

and also

(1.7)

Already Stieltjes considered this kind of functional, see [19, p. 25]. It is also used in [4, Definition 2.1, p.6]).

In the classical case where all the Hankel determinants , these polynomials are proportional to the classical orthonormal polynomials (see [4, p.10; p.15; Exercise 3.1(a), p.17])

(1.8)

and those of the second kind.

In the general case of an arbitrary sequence of real numbers Frobenius [7, (5), p.212] obtained in 1894 a recurrent relation for the polynomials in the following determinant form

(1.9)

where , , and , . If for all , then the functional is called quasi-definite (see [4, Definition 3.2, p.16]) and the monic polynomials

(1.10)

are usually considered for which the recurrence (1.9) is written in the Jacobi form (see [4, Theorem 4.1, p.18])

(1.11)
(1.12)

where the relations (1.12) are invertible (cp. [4, Theorem 4.2, p.19])

(1.13)

and for all . Conversely, given the recurrence formula (1.11) for monic polynomials with two arbitrary real sequences and satisfying for all , we determine by (1.13) and Proposition 1 a quasi-definite functional such that and for all , , by virtue of (1.5), (1.10), (1.6) and (1.7). This fact is known as the generalized Favard theorem for quasi-definite functionals (see [4, Theorem 4.4, p.21]).

By Theorem 1, if is assumed nonzero, i.e., for at least one , there exists such that , and then is a polynomial of degree . In Theorem 2 we derive from the Kronecker identities (2.7) a simple result about zeros of the polynomials and .

In 1881 Kronecker [13] also characterized all those nonzero sequences of real numbers whose Hankel matrices are of finite rank, see Theorem A of Section 2.

In Corollary 1 we provide a new interpretation of this result based on Theorem 3.

The results obtained by Frobenius [7] in 1894 are formulated in Theorem D of Section 3.

In Subsection 5.3 we use Theorem 3 to derive a recent theorem of Berg and Szwarc [2], see Theorem E.

2. Kronecker’s results from 1881

Let and be a nonzero square matrix of order , where being nonzero means that it has at least one nonzero element. For arbitrary and the determinant is called a minor of of order . The largest order of the nonzero minors of is called the rank of the matrix and is denoted by (see [9, p.2]). The rank of an infinite matrix is defined by , where (see [10, p.205]).

Theorem A (Kronecker (1881)).

Let be a nonzero sequence of real numbers, , , , and . Then a necessary and sufficient condition for to have a finite rank is that

(2.1)

The necessity of the condition is formulated in Kronecker [13, p.560], Frobenius [7, p. 204], Gantmacher [10, p.206] and Iohvidov [11, p. 74], while the sufficiency, proved by Kronecker [13, p.563], is less known and can be found in Iohvidov [11, item 11, p.79] .

It has been proved by Kronecker in [13, (), (), p.567]) that if is of finite rank then

(2.2)

Since (1.3) yields , the change of variable in (2.2) shows that it is equivalent to the Taylor expansion at the origin

of the analytic function on the open disk where . Therefore the series in the righthand side of (2.2) converges absolutely for every , and (2.3) below holds by the Cauchy-Hadamard formula (see [17, (2), p.200]).

Conversely, Kronecker proved in [13, p.568] that if the numbers are the coefficients in the expansion (2.5) of for , and then has the rank , provided (see [15, Section 45, p.198], [10, Theorem 8, p.207]). Thus, the following characterization of the Hankel matrices of finite rank holds.

Theorem B.

Let be a nonzero sequence of real numbers and .

(a) If has a finite rank then ,

(2.3)

and

(2.4)

where the series is absolutely convergent for every .

(b) If and there exist , of degree and of degree at most such that

(2.5)

then , where the equality is attained if and have no common roots.

The following theorem of Kronecker [13, pp.560, 561, 571] clarifies the structure of the sequences satisfying (see also [10, Theorem 7, p.205] and [10, p.234]).

Theorem C.

Let be a nonzero sequence of real numbers and .

(a) has a finite rank if and only if and there exist numbers
, , … , such that

(2.6)

(b) If has a finite rank then for every there exist numbers
, , …, such that

where is equal to from (2.6) for each .

(c) If has a finite rank then the sequence is uniquely determined by the values of , , …, .

Finally, we note that the equality (2.4) proved by Kronecker in [13, (), (), p.567]) asserts implicitly that the polynomials and have no common roots provided that . Furthermore, this fact also follows from the identity

(2.7)

written by Kronecker in [13, (F), p.564] for arbitrary (see also [7, (14), p.220]).

Observe that (2.7) can easily be proved when for all (see [8, III.15, p.48], [3, Theorem 2.12, p.54]). These restrictions can be removed by the so-called perturbation technique. More precisely, by using the Hilbert matrix , for every we introduce the perturbed sequence

whose Hankel determinant for every is a polynomial of degree in the variable with positive leading coefficient (see [16, 3, p.92]) . Since the zeros of all polynomials , , form an at most countable set, there exists a sequence of positive numbers tending to zero as such that

With (2.7) in hand for , , we conclude by the continuous dependence of (2.7) on , , that (2.7) holds for .

It also follows from (2.7) that and have no common roots, provided that . We have therefore proved the following property (cf. [3, Theorem 2.14, p.57]).

Theorem 2.

Let be an arbitrary nonzero sequence of real numbers and be a positive integer satisfying . Then , , and the polynomial has no common zeros with the polynomials and .

Observe, that Theorem 2 can also be easily deduced from [6, Theorem 1.9, p.80; Theorem 1.3(ii), p.44]. We will in the sequel use the following notion.

Definition 1.

Let be a nonzero sequence of real numbers. The rank of the infinite Hankel matrix is called the Hankel rank of .

Since , the Hankel rank of a real nonzero sequence can be equal to any positive integer or infinity.

3. Frobenius’ theorem from 1894

Let be an arbitrary nonzero sequence of real numbers and

(3.1)

Theorem 1 yields . Suppose that consists of () distinct elements arranged in increasing order and , where it is assumed that for arbitrary . Then

(3.2)

We say that the Hankel determinant polynomial defined by (1.3) is of full degree if . It follows from (1.3), (1.4), (3.1) and (3.2) that is of full degree if and only if for some , i.e.,

(3.3)

Theorem 2 states that the identities (2.7) proved by Kronecker in 1881 imply that for each the polynomial is preceded by a nonzero polynomial which has no common zeros with and whose degree can be strictly less than .

In 1894 Frobenius established [7, (10), p.210] that for such is proportional with a nonzero real constant of proportionality to the previous polynomial of full degree provided that , i.e., there exists such that

if and (see also [6, Theorem 1.3(ii), p.44]). Furthermore, he proved in [7, (8), p.214] that for the recurrence relations

hold between the monic polynomials

corresponding to the polynomials of full degree, where are nonzero real numbers and is a monic polynomial of degree for every (see also [6, Remark 1.2, p.71]). It is also proved in [7, (9), p.210] that

provided that and (see also [6, Theorem 1.3, p.44]). Thus, the following theorem was proved by Frobenius [7] in 1894 .

Theorem D.

Let be an arbitrary nonzero sequence of real numbers and , and be defined as in (3.1) and (3.2).

For the Hankel determinant polynomials defined by (1.3) the following assertions hold.

(a) If then there exists such that

when and

when .

(b) If , and then there exists such that

when and

when .

(c) If then for the monic polynomials

there exist monic polynomials in

and nonzero real numbers such that

(3.4)

(d) If then and for all .

It should be noted that Theorem D (d) follows directly from Theorem A and Theorem C (a). Indeed, the conditions of Theorem D (d) imply the validity of (2.1) for and in view of Theorem A we obtain that has a finite rank . But for arbitrary the -th row of the determinant for in (1.3) is the linear combination of the first rows by virtue of (2.6). Hence, and the desired result is proved.

Theorem D shows that except of polynomials of full degree and proportional to them the sequence defined in (1.3) contains no other nonzero polynomials. Furthermore, if then it follows from that while and imply . Observe that Theorem D (c) was essentially generalized by A.Draux [6, Theorem 6.2, p.477] in 1983.

4. Iohvidov’s approach from 1969

Throughout this section we fix an arbitrary nonzero sequence of real numbers and use the set defined as in (3.1). The analysis below will not use the statements from the previous Sections 2 and 3.

In 1969 Iohvidov [12] (see also [11]) suggested a new technique for dealing with Hankel matrices. For every he proposed to use the approximating sequence defined as follows.

We first put

(4.1)

Since the first numbers , , , of the sequence satisfy