On Matrix Hurwitz Type Polynomials and Their Interrelations to Stieltjes Positive Definite Sequences and Orthogonal Matrix Polyn

Linear Algebra and its Applications 476 (2015) 56–84

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive deﬁnite sequences and orthogonal matrix polynomials

Abdon Eddy Choque Rivero 1,2

Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, Morelia, Mich., C.P. 58048, Mexico a r t i c l e i n f o a b s t r a c t

Article history: This paper is a direct continuation of the author’s recent inves- Received 13 September 2013 tigations [4] on the non-degenerate truncated matricial Stielt- Accepted 1 March 2015 jes problem. Inspired by a characterization of scalar Hurwitz Available online 16 March 2015 polynomials via continued fractions (see e.g. Gantmacher [15]) Submitted by A. Böttcher a class of matrix Hurwitz type polynomials is introduced. It is Dedicated to my teacher at Leipzig shown that this class is intimately related with Stieltjes pos- University, Professor Bernd Kirstein itive definite sequences of matrices and can be expressed in terms of the associated Stieltjes quadruple of sequences of left MSC: orthogonal matrix polynomials. It is shown that a monic ma- 34E05 trix polynomial is a matrix Hurwitz type polynomial if and 42C05 only if the sequence of its Markov parameters is a Stieltjes 44A60 93D05 positive definite sequence. © 2015 Elsevier Inc. All rights reserved. Keywords: Matrix Hurwitz type polynomials Orthogonal matrix polynomials Matrix continued fractions Markov parameter Stieltjes positive definite sequences of matrices Matricial Stieltjes moment problem Stability theory

E-mail address: [email protected]. 1 Research by this author is partially supported by CONACYT grant No. 153184, CIC–UMSNH, México. 2 The author would like to thank Dr. Conrad Mädler for his valuable comments. http://dx.doi.org/10.1016/j.laa.2015.03.001 0024-3795/© 2015 Elsevier Inc. All rights reserved. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 57

1. Introduction

In the center of this paper stands a certain matrix generalization of the classical notion of Hurwitz polynomials. A real polynomial with all its roots in the open left half plane C− := {z ∈ C: Re z ∈ (−∞, 0)} of the complex plane is called Hurwitz polynomial. Apparently, such polynomials were first studied by J.C. Maxwell [25] in 1868 and later considered by E.J. Routh [28] and A. Hurwitz [19] in 1875 and 1895, respectively. Routh and Hurwitz gave a necessary and sufficient condition for a real polynomial to be a Hurwitz polynomial. Other tools for dealing with such polynomials are provided by Sturm’s theorem [30] and Bezoutian approach [14]. The so-called Markov parameters (MP) approach was studied in [15]; see also [6,18] and references therein. Recently, by using Wall’s continued fractions, a connection between the Schwarz matrices and rational functions associated with Hurwitz polynomials was studied in [31]. In [8] the description of functions generating infinite totally non-negative Hurwitz matrices via Stieltjes meromorphic functions is given. What concerns a detailed treatment of the theory of Hurwitz polynomials we refer the reader to the monographs Gantmacher [15, Chapter XV], Postnikov [26] and Rahman/Schmeisser [27, Section 11.4]. Starting with the classical work of Grommer [17] the theory of Hurwitz polynomials was extended to entire functions (see Chebotarev/Meiman [2], Krein [21], Levin [23, Chapter VII], Katsnelson [20]). The direct matricial generalization of the notion should be defined as follows: A q × q matrix polynomial P is called Hurwitz polynomial if the inclusion {z ∈ C: det P (z) =0} ⊆ C− is satisfied. In this paper, we follow another line of generalizing the classical notion. We are inspired by the investigations in Gantmacher [15, Chapter XV] on Hurwitz polynomials and related mechanical application in Gant- macher/Krein [16, Appendix II]. One of the central ideas there is to study interrelations between the even and odd parts of a polynomial. The membership of a polynomial to the class of Hurwitz polynomials is characterized in terms of particular continued fraction representations of the rational functions which are formed by the even and odd parts of the given polynomial (see Gantmacher [15, Chapter XV, Section 14, Theorem 16]). The shape of these continued fractions immediately establishes a bridge to the Stieltjes moment problem. These connections are discussed in [15, Chapter XV, Section 16]. This paper is based on the author’s recent investigations [4] on the non-degenerate truncated matricial Stieltjes moment problems. In [4] we obtained distinguished matrix continued fraction expansions for the two extremal solutions to the above mentioned moment problem. Against to this background we introduce the notion “matrix Hurwitz type polynomial” which in the scalar case indeed coincides with the classical notion. In the matrix case the relation between “matrix Hurwitz type polynomials” and “matrix Hurwitz polynomials” is still uncovered. A closer look at [4, Theorems 3.4 and 4.8] shows that a Stieltjes positive definite sequence produces matrix continued fractions which lead in a natural way to matrix Hurwitz type polynomials (see Theorem 5.4). Starting from this observation we estab- lish a principle of constructing matrix Hurwitz type polynomials by using the Stieltjes 58 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 quadruple of sequences of left orthogonal matrix polynomials associated with a Stielt- jes positive definite sequence (see Theorem 6.1). Conversely, we show that each matrix Hurwitz type polynomial can be generated in this way (see Theorem 7.9). In this way a bijective correspondence between matrix Hurwitz type polynomials and finite Stieltjes positive definite sequences is established via Markov parameter sequences. In Section 8, we obtain in the scalar case a criterion of asymptotic stability of a system of linear ordinary differential equations with constant coefficients in terms of the main results of this paper (see Theorem 8.2).

2. Matrix Hurwitz type polynomials

We introduce in this section a class of q × q matrix polynomials which turns out to coincide in the classical case q =1with the class of classical Hurwitz polynomials. In our further considerations we will often work with the odd and even parts of a polynomial, which will be introduced now.

Deﬁnition 2.1. Let n ∈ N and let fn be a q × q matrix polynomial of degree n. For z ∈ C let fn be given by

n n−k fn(z)= Akz . (2.1) k=0

Let m ∈ N be chosen such that n =2m or n =2m − 1is satisﬁed. Then the q × q matrix q×q polynomial hn: C → C deﬁned by m A zm−, if n =2m h (z):= =0 2 (2.2) n m m− − =1 A2−1z , if n =2m 1

q×q is called the even part of fn, whereas the q × q matrix polynomial gn: C → C deﬁned by m m− A − z , if n =2m g (z):= =1 2 1 (2.3) n m m− − =1 A2−2z , if n =2m 1 is called the odd part of fn.

Remark 2.2. Let n ∈ N and let fn be a q × q matrix polynomial of degree n. Denote by hn and gn the even and odd part of fn, respectively. Then it is easily checked by straightforward computation that the identity

2 2 fn(z)=hn(z )+zgn(z ) holds for z ∈ C. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 59

Remark 2.3. Let n ∈ N and let fn be a q × q matrix polynomial of degree n. Denote by hn and gn the even and odd part of fn, respectively.

(a) Suppose that n =2m for some m ∈ N. Then the leading coefficients of fn and hn coincide. In particular, fn is monic if and only if hn is monic, and furthermore fn has non-singular leading coefficient if and only if hn has non-singular leading coefficient. (b) Suppose that n =2m − 1for some m ∈ N. Then the leading coefficients of fn and gn coincide. In particular, fn is monic if and only if gn is monic, and furthermore fn has non-singular leading coefficient if and only if gn has non-singular leading coefficient.

Remark 2.4. Let m ∈ N.

(a) Let h be a q × q matrix polynomial of degree m and let g be a q × q matrix polynomial of degree at most m − 1. Let f: C → C be deﬁned by f(z) := h(z2) + zg(z2). Then f is a q × q matrix polynomial of degree 2m with even part h and odd part g. (b) Let h be a q × q matrix polynomial of degree at most m −1and let g be a q × q matrix polynomial of degree m −1. Let f: C → C be deﬁned by f(z) := h(z2) +zg(z2). Then f is a q × q matrix polynomial of degree 2m − 1with even part h and odd part g.

Lemma 2.5. Let n ∈ N and let fn be a q × q matrix polynomial of degree n with non- N { ∈ C } N singular leading coeﬃcient. Let fn := z : det fn(z) =0. Then fn is a ﬁnite subset of C.

∈ C ∨ C → Cq×q Proof.r Fo z let fn be given by (2.1) and let fn : be deﬁned by

n ∨ ∗ k fn (z):= Akz . k=0

∨ ∗ ∨ ∨ Then fn (0) = A0. Hence, det[fn (0)] = det A0 =0. Thus, since det fn is a polynomial, ∨ N ∨ { ∈ C } C ∈ C \{ } the set fn := z : det fn (z) =0 is a ﬁnite subset of . For z 0 the identity

∗ 1 f (z)=zn f ∨ n n z holds (see e.g. [7, Lemma 1.2.2]). Hence,

1 det [f (z)] = znqdet f ∨ . n n z

This implies that det fn does not identically vanish in C. Since det fn is a polynomial the N C 2 set fn is a ﬁnite subset of . 60 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

Lemma 2.6. Let n ∈ N and let fn be a q × q matrix polynomial of degree n with non-singular leading coefficient. Denote by hn and gn the even and odd parts of fn, respectively. Let m ∈ N be chosen such that n =2m or n =2m − 1 is satisfied. If n =2m N { ∈ C } − then the set hn := z : det hn(z) =0 is finite whereas in the case n =2m 1 the N { ∈ C } set gn := z : det gn(z) =0 is finite.

Proof. In the case n =2m (resp. n =2m − 1) the application of Lemma 2.5 in combi- nation with (2.2) (resp. (2.3)) yields the assertion. 2

Now we introduce the central objects of this paper. What concerns the case of an even n we are inspired by a classical result due to Stieltjes [29] and its connections with classical Hurwitz polynomials (see Gantmacher [15, Chapter XV, Section 16, Proposi- tions 15 and 16]). In the case of an odd number n our construction diﬀers from that one which is suggested by the scalar result. The reason for this will be seen later. In view of Lemma 2.6 the following deﬁnition is correct. For A, B ∈ Cq×q with B A −1 invertible, set B := AB .

Deﬁnition 2.7. Let n ∈{2, 3, ...} and let fn be a monic q × q matrix polynomial of degree n. Denote by hn and gn the even and odd parts of fn, respectively.

(a) Suppose n =2m for some m ∈ N. Then fn is called a matrix Hurwitz type polynomial (short MHTP) of degree n if there exists an ordered pair of sequences m−1 m−1 Cq×q [(ck)k=0 , (dk)k=0 ]from > such that the identity

g (z) Iq n = (2.4) hn(z) Iq zc0 + Iq d + 0 . .. dm−2 + −1 zcm−1 + dm−1

∈ C \N m−1 m−1 holds for all z hn . In this case the pair [(ck)k=0 , (dk)k=0 ]is called a Hurwitz parametrization of fn. (b) Suppose n =2m +1 for some m ∈ N. Then fn is called a matrix Hurwitz type polynomial (short MHTP) of degree n if there exists an ordered pair of sequences m m−1 Cq×q [(ck)k=0, (dk)k=0 ]from > such that the identity

h (z) Iq n = (2.5) zgn(z) Iq zc0 + Iq d + 0 . .. zc − + m 1 −1 −1 dm−1 + z cm A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 61

∈ C \ { } ∪N m m−1 holds for all z [ 0 gn ]. In this case the pair [(ck)k=0, (dk)k=0 ]is called a Hurwitz parametrization of fn.

Remark 2.8. Let m ∈ N and let n := 2m +1. Let fn be a MHTP of degree n and let m m−1 [(ck)k=0, (dk)k=0 ]be a Hurwitz parametrization of fn. By multiplying (2.5) by z and taking the inverse of hn(z) , we see that gn(z)

gn(z) Iq = c0 + (2.6) hn(z) Iq zd0 + Iq c + 1 . .. cm−1 + −1 zdm−1 + cm

∈ C \N holds for all z hn .

In the scalar case, i.e. for q =1, the polynomials of Deﬁnition 2.7 coincide, in view of (2.4), (2.6) and [15, Chapter XV, Section 14, Theorem 16], with the classical Hurwitz polynomials. In our subsequent considerations we will use a matricial generalization of the construction of Markov parameters of a polynomial (see Gantmacher [15, Chapter XV, Section 15]). Our approach is based on the following observation.

Lemma 2.9. Let n ∈ N and let fn be a q × q matrix polynomial of degree n with invertible leading coeﬃcient. Denote by hn and gn the even and odd parts of fn, respectively.

∈ N {| | ∈N } (a) Suppose n =2m for some m . Let ηn := max z : z hn . Then there exists a ∞ Cq×q unique sequence (˜sj)j=0 from such that

∞ g (z) n = (−1)jz−(j+1)s˜ h (z) j n j=0

for all z ∈ C with |z| >ηn. ∈ N {| | ∈N } (b) Suppose n =2m +1 for some m . Let ρn := max z : z gn . Then there ∞ Cq×q exists a unique sequence (˜sj)j=0 from such that

∞ h (z) n = (−1)jz−(j+1)s˜ zg (z) j n j=0

for all z ∈ C with |z| >ρn. 62 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

Proof. This follows by observing that in view of (2.2) and (2.3) we have deg hn = m and deg gn = m − 1in the case n =2m, and furthermore deg hn = m and deg gn = m in the case n =2m +1. 2

Lemma 2.9 leads us to the following notion which will be important for our subsequent considerations. What concerns the following definition we meet the same situation as in Definition 2.7. Namely, in the case of even n we have a direct generalization of the classical notion of Markov parameters whereas in the odd case we prefer a different construction.

Deﬁnition 2.10. Let n ∈ N and let fn be a q × q matrix polynomial of degree n with ∞ invertible leading coeﬃcient. Denote by (˜sj)j=0 the sequence from Lemma 2.9. Then n ∞ (˜sj )j=0 is called the sequence of Markov parameters of fn. Furthermore, (˜sj)j=0 is called the extended sequence of Markov parameters of fn

3. On matricial power moment problems and several classes of sequences of matrices

Our subsequent considerations will be closely related to matricial power moment problems. In order to give precise formulations of the moment problems under consideration we introduce some terminology. Let Ωbe a Borelian subset of the real axis R. Further, q let q ∈ N, let BΩ be the σ-algebra of all Borelian subsets of Ω, and let M≥(Ω) be the set of all non-negative Hermitian q × q measures on (Ω, BΩ). Let κ ∈ N0 ∪{∞}and let Mq ∈Mq ≥,κ(Ω) be the set of all σ ≥(Ω) such that the integral (σ) j sj := t σ(dt) Ω exists for all non-negative integers j ≤ κ. Initiated in the scalar case q =1by M.G. Krein [22] the following truncated matrix moment problem is studied:

m ≤ ∈ N m × M;([Ω sj )j=0, ]Let m 0 and let (sj )j=0 be a sequence of complex q q matrices. Mq m ≤ ∈Mq − (σ) Describe the set ≥[Ω; (sj)j=0, ]of all σ ≥,m(Ω) for which sm sm is (σ) non-negative Hermitian and, in the case m > 0, moreover sj = sj is fulﬁlled for all j ∈ Z0,m−1.

There are many interesting interrelations between moment problems on [0, ∞)on R ∈Mq ∞ m ≤ the one side and on on the other side. Obviously each σ ≥[[0, ); (sj)j=0, ] ∈Mq R m ≤ determines by a natural zero continuation a measure σ¯ ≥[ ;(sj )j=0, ]. So the case Ω = R stands always in the background. To discuss the solvability of the above mentioned moment problems, we introduce some ∈ N 2n Cq×q classes of sequences of matrices. Let n 0 and let (sj)j=0 be a sequence from . A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 63

2n Then (sj)j=0 is called Hankel non-negative deﬁnite (resp. Hankel positive deﬁnite) if the block Hankel matrix

n Hn := (sj+k)j,k=0

H≥ H> is non-negative Hermitian (resp. positive Hermitian). Denote by q,2n (resp. q,2n) the 2n set of all Hankel non-negative deﬁnite (resp. Hankel positive deﬁnite) sequences (sj)j=0 from Cq×q.

∈ N 2n ∈H≥ 2n ∈H> ∈ Remark 3.1. Let n and let (sj)j=0 q,2n (resp. (sj)j=0 q,2n). Then for k Z 2k ∈H≥ 2k ∈H> 0,n−1 it is obvious that (sj )j=0 q,2k (resp. (sj)j=0 q,2k).

∞ Cq×q Remark 3.1 leads us to the following notion. A sequence (sj)j=0 from is called Hankel non-negative definite (resp. Hankel positive definite) if for all n ∈ N0 the sequence 2n (sj )j=0 is Hankel non-negative definite (resp. Hankel positive definite). We denote be H≥ H> q,∞ (resp. q,∞) the set of all Hankel non-negative definite (resp. Hankel positive Cq×q H≥ definite) sequences from . The following result shows the importance of the set q,2n.

∈ N 2n Theorem 3.2. (See [3, Theorem 3.2].) Let n 0 and let (sj)j=0 be a sequence of complex × Mq R 2n ≤ ∅ 2n ∈H≥ q q matrices. Then ≥[ ;(sj )j=0, ] = if and only if (sj )j=0 q,2n.

The present work is closely related to the non-degenerate situation of Prob- ∞ m ≤ lem M[[0, ); (sj)j=0, ]. To describe the solvability of this problem we introduce further ∈ N 2n classes of sequences of complex matrices. Let n and let (sj )j=0 be a sequence Cq×q 2n from . Then (sj)j=0 is called Stieltjes non-negative definite (resp. Stieltjes posi- 2n ∈H≥ 2(n−1) ∈H≥ 2n ∈H> tive definite) if (sj)j=0 q,2n and (sj+1)j=0 q,2(n−1) (resp. (sj)j=0 q,2n and 2(n−1) ∈H> 2n+1 Cq×q 2n+1 (sj+1)j=0 q,2(n−1)). Let (sj )j=0 be a sequence from . Then (sj)j=0 is called { 2n 2n } ⊆ Stieltjes non-negative definite (resp. Stieltjes positive definite) if (sj)j=0, (sj+1)j=0 H≥ { 2n 2n } ⊆H> ∈ N K≥ K> q,2n (resp. (sj)j=0, (sj+1)j=0 q,2n). For m the symbol q,m (resp. q,m) stands for the set of all Stieltjes non-negative definite (resp. Stieltjes positive definite) m Cq×q sequences (sj)j=0 from .

∈ N m ∈K≥ m ∈K> ∈ Z Remark 3.3. Let m and let (sj)j=0 q,m (resp. (sj)j=0 q,m). Then for 0,m ∈K≥ ∈K> it is obvious that (sj )j=0 q, (resp. (sj)j=0 q,).

∞ Cq×q Remark 3.3 leads us to the following notion. A sequence (sj)j=0 from is called Stieltjes non-negative definite (resp. Stieltjes positive definite) if for all m ∈ N0 m the sequence (sj )j=0 is Stieltjes non-negative definite (resp. Stieltjes positive defi- K≥ K> nite). We denote be q,∞ (resp. q,∞) the set of all Stieltjes non-negative definite (resp. Stieltjes positive definite) sequences from Cq×q. Concerning a detailed treatment of the theory of Stieltjes non-negative definite sequences we refer the reader to Dyukarev/Fritzsche/Kirstein/Mädler [10], Fritzsche/Kirstein/Mädler [11,12]. 64 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

The following result provides the reasoning why we will work at many places in this paper with infinite Stieltjes positive definite sequences instead of finite ones.

∈ N m ∈ Proposition 3.4. (See [11, Propositions 4.13 and 4.14].) Let m 0 and let (sj)j=0 K> ∞ ∈K> m m q,m. Then there exists a sequence (˜sj)j=0 q,∞ such that (˜sj)j=0 =(sj )j=0.

K≥ The following result shows the importance of the set q,m.

∈ N m Theorem 3.5. (Cf. [10, Theorem 1.4].) Let m and let (sj)j=0 be a sequence of complex × Mq ∞ m ≤ ∅ m ∈K≥ q q matrices. Then ≥[[0, ); (sj )j=0, ] = if and only if (sj )j=0 q,m.

∈ N m ∈K> Remark 3.6. Let m and let (sj )j=0 q,m. Then Theorem 3.5 shows that Mq ∞ m ≤ ∅ ≥[[0, ); (sj)j=0, ] = .

For a detailed description of the work on matrix versions of the Stieltjes moment problem we refer the reader to [12] and the references therein. This paper is intimately related to several aspects of the author’s recent investigations [4] on Prob- ∞ m ≤ m ∈K> lem M[[0, ); (sj)j=0, ]in the case (sj)j=0 q,m. First we explain that, following the classical line, we did not study the original moment problem but an equivalent problem for holomorphic q × q matrix-valued functions in C \ [0, ∞).

q q×q Deﬁnition 3.7. Let σ ∈M≥([0, ∞)). Then the function Gσ: C \ [0, ∞) → C deﬁned by 1 G (z):= σ(dt) σ t − z [0,∞) is called Stieltjes transform of σ.

In view of a matrix version of the Stieltjes–Perron inversion formula (see [5, Theo- q rem 8.2]) a measure σ ∈M≥([0, ∞)) is uniquely determined by its Stieltjes transform. For this reason the solution set of the Stieltjes moment problem can be described in terms of Stieltjes transform s. In this way, Stieltjes [29] already handled the classical case. m ∈K> In the case (sj)j=0 q,m the set of all Stieltjes transforms of the measures ∞ m ≤ belonging to the solution set of Problem M[[0, ); (sj)j=0, ]was parametrized by Yu.M. Dyukarev [9] with the aid of a linear fractional transformation (see also [4, The- orem 3.2]). Of particular importance for our subsequent considerations is the fact that the generating matrix-valued polynomial of this linear fractional transformation was expressed in [4, Theorem 4.6] in terms of a quadruple of q × q matrix polynomials with particular orthogonality properties. In our subsequent considerations we will sometimes meet compactly supported matrix measures on R. In this case the Stieltjes transform has a particularly simple shape. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 65

∈ R ∈Mq ∈Mq Lemma 3.8. Let a, b with a max{|a|, |b|} the representation ∞ − −(j+1) (σ) Sσ(z)= z sj . j=0

Proof. If z ∈ C satisﬁes |z| > max{|a|, |b|} then

∞ ∞ 1 1 1 t k = − = − = − tkz−(k+1) t − z z(1 − t/z) z z k=0 k=0 for all t ∈ [a, b]. Now changing integration and summation yields the assertion. 2

4. The Dyukarev–Stieltjes parametrization of Stieltjes positive deﬁnite sequences

∞ ∈ In this section we recall the Dyukarev–Stieltjes parametrization of sequences (sj)j=0 K> q,∞ which was introduced in Fritzsche/Kirstein/Mädler [12, Section 8] inspired by Yu.M. Dyukarev’s paper [9]. For a complex p × q matrix A, let A∗ be the conjugate transpose of A and let A† be the Moore–Penrose inverse of A, i.e., the unique matrix X ∈ Cq×p which satisﬁes the four equations AXA = A, XAX = X, (AX)∗ = AX, and (XA)∗ = XA. If A ∈ Cq×q, then let det A be the determinant of A. For a complex q × q matrix A with det A =0let A−1 be the inverse of A. In this case, A† = A−1. ∞ Cp×q Let (sj )j=0 be a sequence from . First we introduce some sequences of matrices ∞ associated with (sj )j=0. Let ⎛ ⎞ s0 s1 ... sj ⎜ ⎟ ⎜ s1 s2 ... sj+1 ⎟ H1,j := ⎜ . . . ⎟ ,j∈ N0, (4.1) ⎝ . . . ⎠

sj sj+1 ... s2j ⎛ ⎞ s1 s2 ... sj+1 ⎜ ⎟ ⎜ s2 s3 ... sj+2 ⎟ H2,j := ⎜ . . . ⎟ ,j∈ N0, (4.2) ⎝ . . . ⎠

sj+1 sj+2 ... s2j+1 ⎛ ⎞ sj ⎜ ⎟ ⎜ sj+1 ⎟ Y1,j := ⎜ . ⎟ ,Z1,j := (sj,sj+1,...,s2j−1),j∈ N, (4.3) ⎝ . ⎠

s2j−1 66 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 ⎛ ⎞ sj+1 ⎜ ⎟ ⎜ sj+2 ⎟ Y2,j := ⎜ . ⎟ ,Z2,j := (sj+1,sj+2,...,s2j),j∈ N. (4.4) ⎝ . ⎠

s2j

Furthermore, for j ∈ N0 we set s0, if j =0 H1,j := − † ≥ , (4.5) s2j Z1,j H1,j−1Y1,j, if j 1 s1, if j =0 H2,j := − † ≥ . (4.6) s2j+1 Z2,jH2,j−1Y2,j, if j 1

In other words, if j ∈ N then H1,j (resp. H2,j) is the Schur complement of s2j (resp. s2j+1) in H1,j (resp. H2,j). Now we summarize some basic properties of Hankel positive deﬁnite sequences which are mostly taken from [13, Section 3].

∞ ∈H> Remark 4.1. Let (sj )j=0 q,∞. Then:

∈ N ∈ Cq×q ∈ Cq×q (a) If j 0, then s2j > and s2j+1 H . ∈ N ∈ C(j+1)q×(j+1)q ∈ ∞ (b) If j 0, then H1,j > (and in particular det H1,j (0, + )). ∈ N ∈ Cq×q ∈ ∞ (c) If j 0, then H1,j > (and in particular det H1,j (0, + )).

K> From Remark 4.1 and the deﬁnition of the set q,∞ we immediately obtain the following observations.

∞ ∈K> Remark 4.2. Let (sj )j=0 q,∞. Then:

∈ N ∈ Cq×q (a) If j 0, then sj > . ∈{ } ∈ N ∈ C(j+1)q×(j+1)q ∈ (b) If k 1, 2 and j 0, then Hk,j > (and in particular det Hk,j (0, +∞)). ∈{ } ∈ N ∈ Cq×q ∈ ∞ (c) If k 1, 2 and j 0, then Hk,j > (and in particular det Hk,j (0, + )).

Let Iq v0 := Iq,vk := ,k∈ N. (4.7) 0kq×q

∞ Cp×q ∈ N Let (sj )j=0 be a sequence from . Let j, k 0. Then, we set ⎧ ⎪ 0 × , if j>k ⎪ ⎛q q ⎞ ⎨ sj y := sj+1 (4.8) [j,k] ⎪ ⎝ . ⎠ , if j ≤ k ⎩⎪ . sk A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 67 and 0q×q, if j>k z[j,k] := . (4.9) (sj ,sj+1,...,sk), if j ≤ k

The following construction goes back to Yu.M. Dyukarev [9, p. 77].

∞ ∈K> Deﬁnition 4.3. (See [12, Deﬁnition 8.2].) Let (sj )j=0 q,∞ and let −1 s0 , if k =0 Mk := ∗ −1 − ∗ −1 ≥ vkH1,k vk vk−1H1,k−1vk−1, if k 1 and −1 s0s1 s0, if k =0 Lk := ∗ −1 − ∗ −1 ≥ y[0,k]H2,k y[0,k] y[0,k−1]H2,k−1y[0,k−1], if k 1

∈ N ∞ ∞ for all k 0. Then the ordered pair [(Lk)k=0, (Mk)k=0]is called the Dyukarev–Stieltjes ∞ parametrization (shortly DS-parametrization) of (sj )j=0.

∞ ∈K> Remark 4.4. (See [12, Remark 8.23].) Let (sj)j=0 q,∞ with DS-parametrization ∞ ∞ [(Lk)k=0, (Mk)k=0]. Then, in view of [9, Theorem 7], the matrices Lk and Mk are positive Hermitian and, in particular, invertible for all k ∈ N0.

∞ ∈K> Remark 4.5. (See [12, Remark 8.3].) Let (sj)j=0 q,∞ with DS-parametrization ∞ ∞ [(Lk)k=0, (Mk)k=0]. In view of Deﬁnition 4.3, (4.7), (4.1), (4.8), and (4.2), we can easily see then, that, for all k ∈ N0, the matrix Mk only depends on the matrices s0, ..., s2k and that the matrix Lk only depends on the matrices s0, s1, ..., s2k+1.

Against to the background of Remark 4.5 we see that the notion “DS-parametrization” could also be analogously introduced for ﬁnite Stieltjes positive deﬁnite sequences. Re- mark 4.5 leads us to the following notion.

∈ N n ∈K> ∞ ∈K> Definition 4.6. Let n and let (sj)j=0 q,n. Let (˜sj)j=0 q,∞ be chosen such that n n ∞ ∞ (˜sj )j=0 =(sj)j=0 is satisfied and let [(Lk)k=0, (Mk)k=0]be the DS-parametrization ∞ ∈ N − of (˜sj)j=0. Let m be chosen such that n =2m or n =2m 1is satisfied. m−1 m m−1 m−1 Then the ordered pairs [(Lk)k=0 , (Mk)k=0]and [(Lk)k=0 , (Mk)k=0 ]are called the DS- 2m 2m−1 parametrization of (sj)j=0 and (sj)j=0 , respectively.

∈ N n ∈K> ∈ N Remark 4.7. Let n and let (sj)j=0 q,n. Let m be chosen such that n =2m − m−1 m m−1 m−1 or n =2m 1is satisﬁed. Denote by [(Lk)k=0 , (Mk)k=0](resp. [(Lk)k=0 , (Mk)k=0 ]) n the DS-parametrization of (sj)j=0. Then, in view of Deﬁnition 4.6 and Remark 4.4, the matrices Lk and M are positive Hermitian and, in particular, invertible for all k ∈ Z0,m−1 and all ∈ Z0,m (resp. ∈ Z0,m−1). 68 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

The following proposition plays a key role in the proof of Theorem 7.9 which is the second main result of present work. This proposition is a slightly modified version of the Fritzsche/Kirstein/Mädler’s Proposition 8.27 in [12], which considers an infinite sequence ∞ ∞ of pairs [(Lk)k=0, (Mk)k=0]instead of an analogue finite sequence. Our proof differs from the one in [12] since it is based on part d) of Theorem 4.12 in [11] which in turn uses the rank of the matrices H1,j and H2,j. Let us first recall the following well-known result (see e.g. [7, Lemma 1.1.9] or [1, Proposition 8.2.4]). A11 A12 Lemma 4.8. Let A := ∗ be a Hermitian (n + m) × (n + m) matrix with A12 A22 n × n block A11. Then the following statements are equivalent:

∈ C(n+m)×(n+m) (i) A > . ∈ Cn×n − ∗ −1 ∈ Cm×m (ii) A11 > and A22 A12A11 A12 > . ∈ Cm×m − −1 ∗ ∈ Cn×n (iii) A22 > and A11 A12A22 A12 > .

∈ N m−1 m m m Proposition 4.9. Let m and let (Lj)j=0 and (Mj)j=0 (resp. (Lj)j=0 and (Mj)j=0) × 2m be two sequences of positive Hermitian complex q q matrices. Let the sequence (sj)j=0 2m+1 (resp. (sj )j=0 ) be recursively deﬁned by −1 M0 , if j =0 − − s2j := →j 1 →j 1 (4.10) ∗ −1 −∗ −1 −1 ≥ Y1,jH1,j−1Y1,j +( k=0 MkLk) Mj ( k=0 MkLk) , if j 1 and −∗ −1 (M0L0) L0(M0L0) , if j =0 → → s2j+1 := j j (4.11) ∗ −1 −∗ −1 ≥ Y2,jH2,j−1Y2,j +( k=0 MkLk) Lj( k=0 MkLk) , if j 1

2m 2m+1 for j =0, ..., 2m (resp. j =0, ..., 2m +1). Then (sj )j=0 (resp. (sj)j=0 ) is a Stieltjes positive deﬁnite sequence.

Proof. From (4.10) and (4.5), we obtain

⎛ ⎞−∗ ⎛ ⎞− j−1 j−1 1 → → −1 ⎜ ⎟ −1 ⎜ ⎟ H1,0 = M0 , H1,j = ⎝ MkLk⎠ Mj ⎝ MkLk⎠ . (4.12) k=0 k=0

Respectively from (4.11) and (4.6) we get

⎛ ⎞−∗ ⎛ ⎞− j j 1 → → −1 −1 −1 ⎜ ⎟ ⎜ ⎟ H2,0 = M0 L0 M0 , H2,j = ⎝ MkLk⎠ Lj ⎝ MkLk⎠ . (4.13) k=0 k=0 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 69

∈ Cq×q ∈ Cq×q Since Lj > and Mj > for all j, then

⎛ ⎞−∗ ⎛ ⎞− j−1 j−1 1 → → ⎜ ⎟ ⎜ ⎟ −1 ∈ Cq×q −1 ∈ Cq×q M0 > , ⎝ MkLk⎠ Mj ⎝ MkLk⎠ > , (4.14) k=0 k=0

⎛ ⎞−∗ ⎛ ⎞− j j 1 → → ⎜ ⎟ ⎜ ⎟ −1 −1 −1 ∈ Cq×q ∈ Cq×q M0 L0 M0 > , ⎝ MkLk⎠ Lj ⎝ MkLk⎠ > . (4.15) k=0 k=0

By (4.12), (4.14) (resp. (4.13), (4.15)), we have

∈ Cq×q ∈ N H1,j > , for all j 0, (4.16) ∈ Cq×q ∈ N H2,j > , for all j 0. (4.17)

Let H1,j−1 Y1,j H2,j−2 Y2,j−1 H1,j = ∗ ,H2,j−1 = ∗ . (4.18) Y1,j s2j Y2,j−1 s2j−1

By using (4.16), (4.18) (resp. (4.17), (4.18)) and Lemma 4.8, we obtain H1,m which is positive Hermitian; (resp. H2,m is positive Hermitian). Consequently, the sequence 2m 2 (sj )j=0 is a Stieltjes positive deﬁnite sequence.

Note that equalities (4.12) and (4.13) were first proved in a different way in [4, Corol- m lary 4.10] under the condition that (sj )j=0 for m =2n and m =2n +1 is Stieltjes positive definite. This result, proved using a different approach, later appears in part (a) of Proposition 8.28 in [12].

5. On the Stieltjes quadruple of sequences of left orthogonal matrix polynomials ∞ ∈K> associated with a sequence (sj )j=0 q,∞

∞ ∈K> Let (sj )j=0 q,∞. Then we will recall the Stieltjes quadruple of sequences of left or- ∞ thogonal matrix polynomials (or short SQSLOMP) associated with (sj )j=0. This notion originates in [4, Deﬁnition 4.1]. For this reason, ﬁrst we introduce some notations. Let 0q×jq 0q×q T0 := 0q×q,Tj := ,j∈ N. (5.1) Ijq 0jq×q

From (5.1) it follows immediately that for z ∈ C we have det(I(j+1)q − zTj) =1. Thus, (j+1)q×(j+1)q the function Rj: C → C given by

−1 Rj(z):=(I(j+1)q − zTj) (5.2) 70 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

is well-deﬁned. Observe, that the matrix-valued function Rj is given for z ∈ C via ⎛ ⎞ Iq 0q×q 0q×q ... 0q×q 0q×q ⎜ ⎟ ⎜ zIq Iq 0q×q ... 0q×q 0q×q ⎟ ⎜ 2 ⎟ ⎜ z Iq zIq Iq ... 0q×q 0q×q ⎟ Rj(z)=⎜ ⎟ . (5.3) ⎝ ...... ⎠ ...... j j−1 j−2 z Iq z Iq z Iq ... zIq Iq

∞ Cq×q Let (sj )j=0 be a sequence from . Using (4.8) we set 0q×q u1,0 := 0q×q,u1,j := ,j∈ N, (5.4) −y[0,j−1] and

u2,j := −y[0,j],j∈ N. (5.5)

∞ ∈K> ∈ C Deﬁnition 5.1. Let (sj )j=0 q,∞. Let z . Let

P1,0(z):=Iq,Q1,0(z):=0q×q,P2,0(z):=Iq,Q2,0(z):=s0. (5.6)

For j ∈ N, let

− ∗ −1 P1,j(z):=( Y1,jH1,j−1,Iq)Rj(z)vj , (5.7) − − ∗ −1 Q1,j (z):= ( Y1,jH1,j−1,Iq)Rj(z)u1,j , (5.8) − ∗ −1 P2,j(z):=( Y2,jH2,j−1,Iq)Rj(z)vj , (5.9) − − ∗ −1 Q2,j (z):= ( Y2,jH2,j−1,Iq)Rj(z)u2,j . (5.10)

∞ ∞ ∞ ∞ Then [(P1,j)j=0, (Q1,j)j=0, (P2,j)j=0, (Q2,j )j=0]is called the Stieltjes quadruple of sequences of left orthogonal matrix polynomials (or short SQSLOMP) associated with ∞ (sj )j=0.

The reasoning for the terminology chosen in Deﬁnition 5.1 was given in [4, Proposi- tion 4.2]. From Deﬁnition 5.1 we immediately see the following observation.

∞ ∈K> ∞ ∞ ∞ ∞ Remark 5.2. Let (sj )j=0 q,∞ and let [(P1,j)j=0, (Q1,j)j=0, (P2,j)j=0, (Q2,j)j=0]be the associated SQSLOMP.

(a) For all j ∈ N0, the matrix polynomials P1,j and P2,j are monic of degree j. (b) For all j ∈ N0, the matrix polynomials Q1,j+1 and Q2,j are of degree j. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 71

∞ ∈K> ∞ ∞ ∞ ∞ Remark 5.3. Let (sj)j=0 1,∞ and let [(P1,j)j=0, (Q1,j)j=0, (P2,j)j=0, (Q2,j )j=0]be the associated SQSLOMP. Then using part (a) of Remark 4.2 it is immediately seen that for k ∈{1, 2} and j ∈ N0 the polynomials Pk,j and Qk,j have real coeﬃcients.

∞ ∈K> The role of the SQSLOMP associated with a sequence (sj)j=0 q,∞ was detailed studied in [4]. These polynomials have particular orthogonality properties (see [4, Propo- sition 4.2]) and can be used to describe the Stieltjes transforms of the measures of the Mq ∞ m ≤ ∈ N solution sets ≥[[0, ); (sj)j=0, ]for all m (see [4, Theorem 4.7]). m ∈K> Mq ∞ m ≤ In the case (sj)j=0 q,m the set ≥[[0, ); (sj)j=0, ]contains two distinguished elements σm,min and σm,max (see [4, formulas (3.12) and (3.13)] for the Stieltjes transforms of these measures). These Stieltjes transforms admit particular matrix continued fraction expansions, which will be recalled now. The following result establishes the bridge between [4] and the present paper.

∞ ∈K> Theorem 5.4. (Cf. [4, Theorems 3.4 and 4.8].) Let (sj)j=0 q,∞ with DS-para- ∞ ∞ ∞ ∞ ∞ ∞ metrization [(Lk)k=0, (Mk)k=0]. Denote by [(P1,j)j=0, (Q1,j )j=0, (P2,j)j=0, (Q2,j)j=0] the ∞ ∈ N ∈ C \ ∞ SQSLOMP associated with (sj)j=0. Further let n 0. For z [0, ), then

∗ I − [Q2,n(z)] q Gσ2n,min (z)= ∗ = z[P2,n(z)] Iq −zM0 + Iq L + 0 . .. + Iq −zM − + n 1 −1 −1 Ln−1 − z Mn and

∗ I −[Q1,n(z)] q Gσ2n,max (z)= ∗ = . [P1,n(z)] Iq −zM0 + Iq L + 0 . .. + Iq L − + n 2 − −1 zMn−1 + Ln−1 A closer look at Theorem 5.4 shows now that a Stieltjes positive definite sequence produces matrix continued fractions of the type occurring in the construction of matrix Hurwitz type polynomials (see Definition 2.7). Furthermore, Theorem 5.4 indicates why we have chosen in the definitions of matrix Hurwitz type polynomials (see Definition 2.7) and of Markov parameters (see Definition 2.10) a version which slightly differs from the classical version. This is caused by the structure of the Stieltjes transform of the extremal solution σn,min of the non-degenerate truncated matricial Stieltjes moment problem. 72 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

6. Construction of matrix Hurwitz type polynomials on the basis of Stieltjes positive deﬁnite sequences

The direction of the investigations of this section is determined by Theorem 5.4 which provides us a principle of constructing matrix Hurwitz type polynomials with the aid of a Stieltjes positive deﬁnite sequence.

∞ ∈K> ∞ ∞ Theorem 6.1. Let (sj)j=0 q,∞ with DS-parametrization [(Lk)k=0, (Mk)k=0]. Denote ∞ ∞ ∞ ∞ ∞ by [(P1,j)j=0, (Q1,j )j=0, (P2,j)j=0, (Q2,j)j=0] the SQSLOMP associated with (sj)j=0. Let m ∈ N.

q×q t(a) Le f2m: C → C be deﬁned by

∗ ∗ m 2 m+1 2 f2m(z):=(−1) P1,m(−z ) + z(−1) Q1,m(−z ) .

(a1) f2m is a MHTP of degree 2m. m−1 m−1 (a2) [(Mk)k=0 , (Lk)k=0 ] is a Hurwitz parametrization of f2m. ∞ (a3) Denote by (sj)j=0 the extended sequence of Markov parameters of f2m. Then sj =˜sj for all j ∈ Z0,2m−1. q×q t(b) Le f2m+1: C → C be deﬁned by

∗ ∗ m 2 m 2 f2m+1(z):=(−1) Q2,m(−z ) + z(−1) P2,m(−z ) .

(b1) f2m+1 is a MHTP of degree 2m +1. m m−1 (b2) [(Mk)k=0, (Lk)k=0 ] is a Hurwitz parametrization of f2m+1. ∞ (b3) Denote by (˜sj)j=0 the extended sequence of Markov parameters of f2m+1. Then sj =˜sj for all j ∈ Z0,2m.

Proof. (a) From Remark 5.2 we can conclude that f2m is a monic q × q matrix polynomial of degree 2m. Furthermore, in view of Remark 5.2 and part (a) of Remark 2.4, m ∗ the even and odd parts h2m and g2m of f2m fulﬁll h2m(z) =(−1) [P1,m(−z)] and m+1 ∗ g2m(z) =(−1) [Q1,m(−z)] for all z ∈ C. Taking additionally into account Theo- m−1 m−1 rem 5.4, we see that f2m is a MHTP of degree 2m and that [(Mk)k=0 , (Lk)k=0 ]is a Hurwitz parametrization of f2m. In view of Theorem 5.4 and Deﬁnitions 2.7 and 2.10, we can conclude

∞ [Q (z)]∗ g (−z) G (z)=− 1,m = 2m = − z−(j+1)s˜ σ2m,max [P (z)]∗ h (−z) j 1,m 2m j=0

(b) From Remark 5.2 we can conclude that f2m+1 is a monic q × q matrix polynomial of degree 2m +1. Furthermore, in view of Remark 5.2 and part (b) of Remark 2.4, the m ∗ even and odd part s h2m+1 and g2m+1 of f2m+1 fulﬁll h2m+1(z) =(−1) [Q2,m(−z)] m ∗ and g2m+1(z) =(−1) [P2,m(−z)] for all z ∈ C. Taking additionally into account Theo- m m−1 rem 5.4, we see that f2m+1 is a MHTP of degree 2m +1 and that [(Mk)k=0, (Lk)k=0 ]is a Hurwitz parametrization of f2m+1. In view of Theorem 5.4 and Deﬁnitions 2.7 and 2.10, we can conclude

∞ [Q (z)]∗ h (−z) G (z)=− 2,m = 2m+1 = − z−(j+1)s˜ σ2m,min z[P (z)]∗ (−z)g (−z) j 2,m 2m+1 j=0

∈ C \ ∞ | | {| | ∈N } for all z [0, )with z >ρ2m+1, where ρ2m+1 := max z : z g2m+1 . Mq ∞ 2m+1 ≤ Since σ2m,min belongs to ≥[[0, ); (sj)j=0 , ]and is concentrated on a ﬁnite set of points and hence compactly supported, we can via Lemma 3.8 conclude sj =˜sj for all j ∈ Z0,2m. 2

Theorem 6.1 leads us to the following notion.

∞ ∈K> ∈{ } Deﬁnition 6.2. Let (sj)j=0 q,∞, let n 2, 3, ... , and let fn be deﬁned as in Theo- ∞ rem 6.1. Then fn is called the MHTP of degree n associated with (sj)j=0.

∞ ∈K> ∈{ } Remark 6.3. Let (sj)j=0 q,∞, let n 2, 3, ... , and let fn be the MHTP of degree n ∞ n−1 associated with (sj )j=0. In view of (5.7)–(5.10), then fn only depends on (sj)j=0 .

Remark 6.3 leads us to the following notion.

∈{ } n−1 ∈K> ∞ ∈K> Deﬁnition 6.4. Let n 2, 3, ... and let (sj)j=0 q,n−1. Let (˜sj )j=0 q,∞ be chosen n−1 n−1 n−1 such that (˜sj)j=0 =(sj)j=0 . Then by the MHTP of degree n associated with (sj)j=0 we ∞ mean the MHTP of degree n associated with (˜sj )j=0.

n−1 7. On associating a Stieltjes positive deﬁnite sequence (sj )j=0 with a matrix Hurwitz type polynomial of degree n

Deﬁnition 6.4 leads us to the following inverse problem. Let n ∈{2, 3, ...} and let fn be a matrix Hurwitz type polynomial of degree n. Then we want to show that there exists n−1 ∈K> a unique sequence (sj)j=0 q,n−1 such that fn coincides with the matrix Hurwitz type n−1 n−1 polynomial of degree n associated with (sj)j=0 . We will show that the sequence (sj)j=0 is given by the sequence of Markov parameters of fn. 74 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

∈ N ∪{∞} κ κ First we introduce some notations. Let κ 0 and let (sj)j=0 and (Ak)k=0 be sequences of complex q × q matrices. For all j ∈ Z0,κ, let ⎛ ⎞ s0 s1 ... sj−1 sj ⎜ ⎟ ⎜ 0q×q s0 ... sj−2 sj−1 ⎟ ⎜ ⎟ ⎜ . .. . . ⎟ S[0,j] := ⎜ . . . . ⎟ . (7.1) ⎝ ⎠ 0q×q 0q×q s0 s1 0q×q 0q×q ... 0q×q s0

Furthermore, let ⎛ ⎞ A2j ⎜ ⎟ ⎜ A2j−2 ⎟ ⎜ ⎟ ⎜ . ⎟ A[j,k] := ⎜ . ⎟ (7.2) ⎝ ⎠ A2(k+1) A2k for all j, k ∈ N0 with k ≤ j and 2j ≤ κ, and let ⎛ ⎞ A2j+1 ⎜ ⎟ ⎜ A2j−1 ⎟ ⎜ ⎟ [j,k] ⎜ . ⎟ A := ⎜ . ⎟ (7.3) ⎝ ⎠ A2k+3 A2k+1 for all j, k ∈ N0 with k ≤ j and 2j +1 ≤ κ. For all m ∈ N let m−1 m−2 2 1 0 Jm := diag (−1) Iq, (−1) Iq,...,(−1) Iq, (−1) Iq, (−1) Iq . (7.4)

Observe that m−1 −Jm−1 0(m−1)q×q (−1) Iq 0q×(m−1)q Jm = = (7.5) 0q×(m−1)q Iq 0(m−1)q×q Jm−1 for all m ∈{2, 3, ...}.

Lemma 7.1. Let n ∈{2, 3, ...} and let fn be a monic q × q matrix polynomial of degree ∈ C ∞ n given for all z by (2.1) with extended sequence of Markov parameters (sj)j=0. Let m ∈ N be chosen such that n =2m or n =2m +1 is satisﬁed. Then

[m−1,0] A = JmS[0,m−1]JmA[m−1,0], (7.6)

Y1,m = H1,m−1JmA[m,1] (7.7) A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 75 if n =2m, and

[m,0] A = Jm+1S[0,m]Jm+1A[m,0], (7.8)

Y2,m = H2,m−1JmA[m,1] (7.9) if n =2m +1.

Proof. Denote by hn and gn the even and odd parts of fn, respectively. Suppose n =2m {| | ∈N } and let ηn := max z : z hn . In view of Deﬁnition 2.10 and part (a) of Lemma 2.9, ∞ gn(z) j −(j+1) then = (−1) z s˜j for all z ∈ C with |z| >ηn. Multiplying both sides by hn(z) j=0 hn(z)results in s s s g (z)= 0 − 1 + ···+ 2m −··· h (z) n z z2 z2m+1 n for all z ∈ C with |z| >ηn. By equating coeﬃcients of equal powers of z on both sides of this identity, we get ⎧ ⎪ A = s A , ⎪ 1 0 0 ⎨⎪ A3 = s0A2 − s1A0, ⎪ . (7.10) ⎪ . ⎪ . ⎩ m−1 A2m−1 = s0A2m−2 − s1A2m−4 + ···+(−1) sm−1A0, and

m 0q×q = skA2m − sk+1A2m−2 + ···+(−1) sk+mA0, for all k ∈ N0. (7.11)

By rewriting (7.10) and (7.11) in matrix form, where k runs from 0to m − 1, yields ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − m−1 A2m−1 s0 s1 ... ( 1) sm−1 A2m−2 ⎜ ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ A2m−3 ⎟ ⎜ . .. . ⎟ ⎜ A2m−4 ⎟ ⎜ . ⎟ = ⎜ ⎟ ⎜ . ⎟ (7.12) ⎝ . ⎠ ⎝ .. ⎠ ⎝ . ⎠ . 0q×q . −s1 . A1 0q×q 0q×q ... s0 A0 and ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ m−1 − − smA0 s0 s1 ... sm 1 ( 1) A2m ⎜ ⎟ ⎜ m−2 ⎟ ⎜ ⎟ s s ... s − (−1) A − ⎜ sm+1A0 ⎟ ⎜ 1 0 m 2 ⎟ ⎜ 2m 2 ⎟ ⎝ ⎠ = ⎜ . . . ⎟ ⎜ . ⎟ . (7.13) ... ⎝ . . . ⎠ ⎝ . ⎠ − s2m 1A0 sm−1 sm ... s2m−2 A2

From (7.3), (7.12), (7.4), (7.1), and (7.2), we obtain (7.6). Taking into account A0 = Iq we can conclude (7.7) from (4.3), (7.13), (4.1), (7.4), and (7.2). In the case n =2m +1 one proves (7.8) and (7.9) in a similar way. 2 76 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

∈{ } n × Remark 7.2. Let n 2, 3, ... and let (Ak)k=0 be a sequence of complex q q matrices. In view of (7.5), (7.2), and (7.3), then m −JmA[m,1] (−1) A2m Jm+1A[m,0] = = (7.14) A0 JmA[m−1,0] for all m ∈ N with 2m ≤ n, and − [m,1] − m [m,0] JmA ( 1) A2m+1 Jm+1A = = [m−1,0] (7.15) A1 JmA for all m ∈ N with 2m +1 ≤ n.

Remark 7.3. Let m ∈ N. According to (7.5) and (5.1), then

∗ − ∗ TmJm+1 = Jm+1Tm, (7.16) which, in view of (5.2), implies

∗ ∗ − Rm(z)Jm+1 = Jm+1Rm( z) (7.17) for all z ∈ C.

∈ N m × Remark 7.4. Let m and let (sj)j=0 be a sequence of complex q q matrices. Using (5.4), (5.5), (5.3), (4.7), (5.1), and (7.1), one can prove by direct calculations that

∗ ∗ − ∗ ∗ ∗ u1,mRm(z)= vmRm(z)TmS[0,m] (7.18) and

∗ ∗ − ∗ ∗ u2,mRm(z)= vmRm(z)S[0,m] (7.19) for all z ∈ C.

∈ N m 2m × Lemma 7.5. Let m , and let (sj)j=0 and (Ak)k=0 be sequences of complex q q matrices. For all z ∈ C, then

∗ ∗ ∗ ∗ ∗ vmTmRm(z)Jm+1S[0,m]Jm+1A[m,0] = vm−1Rm−1(z)JmS[0,m−1]JmA[m−1,0]. (7.20)

Proof. This is proved by using (4.8), the second equality of (7.5), (7.14) and S[0,m] = s y∗ 0 [1,m] . 2 0qm×q S[0,m−1] A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 77

Lemma 7.6. Let n ∈{2, 3, ...} and let fn be a monic q × q matrix polynomial of degree ∈ C ∞ n given for all z by (2.1) with extended sequence of Markov parameters (sj)j=0. Let m ∈ N be chosen such that n =2m or n =2m +1 is satisﬁed. Then − −1 H1,m−1Y1,m Jm+1A[m,0] = (7.21) Iq if n =2m, and − −1 H2,m−1Y2,m Jm+1A[m,0] = (7.22) Iq if n =2m +1.

Proof. In view of A0 = Iq, Eqs. (7.21) and (7.22) follow from (7.5), (7.2), (7.7) and (7.5), (7.2), (7.9), respectively. 2

∞ ∈K> ∞ ∞ ∞ ∞ Lemma 7.7. Let (sj )j=0 q,∞ and let [(P1,j)j=0, (Q1,j )j=0, (P2,j)j=0, (Q2,j)j=0] be the associated SQSLOMP. For all j ∈ N0 and all z ∈ C, then − −1 ∗ ∗ ∗ ∗ H1,j−1Y1,j Q1,j (z)=vj Rj (z)Tj S[0,j] (7.23) Iq and − −1 ∗ ∗ ∗ H2,j−1Y2,j Q2,j (z)=vj Rj (z)S[0,j] . (7.24) Iq

Proof. The identities (7.23) and (7.24) are proved by employing (5.8), (7.18) and (5.10), (7.19), respectively. 2

Proposition 7.8. Let n ∈{2, 3, ...} and let fn be a monic q × q matrix polynomial of n n−1 ∈K> degree n with sequence of Markov parameters (˜sj)j=0 such that (˜sj)j=0 q,n−1. Then n−1 fn is the MHTP of degree n associated with (˜sj)j=0 .

Proof.r Fo z ∈ C let fn be given by (2.1). Denote by hn and gn the even and odd part ∞ ∈K> of fn, respectively. In view of Proposition 3.4, there exists a sequence (sj)j=0 q,∞ with

sj =˜sj for all j ∈ Z0,n−1. (7.25)

∞ ∞ ∞ ∞ Denote by [(P1,j)j=0, (Q1,j )j=0, (P2,j)j=0, (Q2,j)j=0]the SQSLOMP associated with ∞ ∈ N − (sj )j=0. Let m be chosen such that n =2m or n =2m 1is satisﬁed. Then the matrices H1,m−1 and H2,m−1 are both invertible, according to part (b) of Remark 4.2. Let z ∈ C. 78 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

Suppose n =2m. We show that

− m ∗ − − m+1 ∗ − hn(z)=( 1) P1,m( z)andgn(z)=( 1) Q1,m( z).

To this end, ⎛ ⎞ A2m ⎜ ⎟ A2m−2 m ⎜ ⎟ ∗ ∗ ∗ ∗ hn(z)=(Iq,zIq,...,z Iq) ⎜ . ⎟ = vmRm(z)A[m,0] = vmRm(z)Jm+1Jm+1A[m,0] ⎝ . ⎠ A 0 − − −1 ∗ ∗ − JmA[m,1] − m ∗ ∗ − H1,m−1Y1,m = vmJm+1Rm( z) =( 1) vmRm( z) A0 Iq − m ∗ − =( 1) P1,m( z), where the 1st equality is due to (2.2), the 2nd equality is due to (4.7), (5.3) and (7.2), the 3rd equality is due to (7.4), the 4th equality is due to (7.17) and (7.14), the 5th equality is due to (4.7), (7.5), (7.25), (7.7) and A0 = Iq, and the last equality is due to (5.7). Analogously, ⎛ ⎞ A2m−1 ⎜ ⎟ A2m−3 m−1 ⎜ ⎟ ∗ ∗ [m−1,0] gn(z)=(Iq,zIq,...,z Iq) ⎜ . ⎟ = vm−1Rm−1(z)A ⎝ . ⎠

A1 ∗ ∗ ∗ ∗ ∗ = vm−1Rm−1(z)JmS[0,m−1]JmA[m−1,0] = vmTmRm(z)Jm+1S[0,m]Jm+1A[m,0] ∗ ∗ ∗ ∗ ∗ ∗ = v T Jm+1R (−z)S[0,m]Jm+1A[m,0] = −v Jm+1T R (−z)S[0,m]Jm+1A[m,0] m m m m m m −H−1 Y = −(−1)mv∗ T ∗ R∗ (−z)S 1,m−1 1,m m m m [0,m] I q − −1 − m+1 ∗ ∗ − ∗ H1,m−1Y1,m − m+1 ∗ − =( 1) vmRm( z)TmS[0,m] =( 1) Q1,m( z), Iq where the 1st equality is due to (2.3), the 2nd equality is due to (4.7), (5.3) and (7.3), the 3rd equality is due to (7.25) and (7.6), the 4th equality is due to (7.20), the 5th equality is due to (7.17), the 6th equality is due to (7.16), the 7th equality is due to (4.7), (7.5), (7.25) and (7.21), the 8th equality is due to (5.2), and the last equality is due to (7.23). Hence,

− m ∗ − 2 − m+1 ∗ − 2 fn(z)=( 1) P1,m( z )+z( 1) Q1,m( z ).

Since, in view of (5.7) and (5.8), the matrix polynomials P1,m and Q1,m only depend on the matrices s0, s1, ..., s2m−1, which by (7.25) coincide with the matrices n−1 s˜0, s˜1, ..., s˜2m−1, this shows that fn is the MHTP of degree n associated with (˜sj )j=0 . A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 79

Suppose n =2m +1. We show that

− m ∗ − − m ∗ − hn(z)=( 1) Q2,m( z)andgn(z)=( 1) P2,m( z).

To this end, ⎛ ⎞ A2m+1 ⎜ ⎟ A2m−1 m ⎜ ⎟ ∗ ∗ [m,0] hn(z)=(Iq,zIq,...,z Iq) ⎜ . ⎟ = vmRm(z)A ⎝ . ⎠

A1 ∗ ∗ ∗ ∗ = v R (z)Jm+1S[0,m]Jm+1A[m,0] = v Jm+1R (−z)S[0,m]Jm+1A[m,0] m m m m − −1 − m ∗ ∗ − H2,m−1Y2,m − m ∗ − =( 1) vmRm( z)S[0,m] =( 1) Q2,m( z), Iq where the 1st equality is due to (2.2), the 2nd equality is due to (4.7), (5.3) and (7.3), the 3rd equality is due to (7.25) and (7.8), the 4th equality is due to (7.17), the 5th equality is due to (4.7), (7.5), (7.25) and (7.22), and the last equality is due to (7.24). Analogously, ⎛ ⎞ A2m ⎜ ⎟ A2m−2 m ⎜ ⎟ ∗ ∗ ∗ ∗ gn(z)=(Iq,zIq,...,z Iq) ⎜ . ⎟ = vmRm(z)A[m,0] = vmRm(z)Jm+1Jm+1A[m,0] ⎝ . ⎠ A 0 − − −1 ∗ ∗ − JmA[m,1] − m ∗ ∗ − H2,m−1Y2,m = vmJm+1Rm( z) =( 1) vmRm( z) A0 Iq − m ∗ − =( 1) P2,m( z), where the 1st equality is due to (2.3), the 2nd equality is due to (4.7), (5.3) and (7.2), the 3rd equality is due to (7.4), the 4th equality is due to (7.17) and (7.14), the 5th equality is due to (4.7), (7.5), (7.25), (7.9) and A0 = Iq, and the last equality is due to (5.9). Hence,

− m ∗ − 2 − m ∗ − 2 fn(z)=( 1) Q2,m( z )+z( 1) P2,m( z ).

Since, in view of (5.10) and (5.9), the matrix polynomials Q2,m and P2,m only depend on the matrices s0, s1, ..., s2m, which by (7.25) coincide with the matrices s˜0, s˜1, ..., s˜2m, n−1 2 this shows that fn is the MHTP of degree n associated with (˜sj )j=0 .

Now we state the main result of this section.

Theorem 7.9. Let n ∈{2, 3, ...} and let fn be a MHTP of degree n with sequence of n ∈ N Markov parameters (˜sj)j=0. Let m be chosen such that n =2m or n =2m +1 is satisﬁed. 80 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

m−1 m−1 (a) Suppose n =2m and let [(ck)k=0 , (dk)k=0 ] be a Hurwitz parametrization of fn. n−1 ∈K> m−1 m−1 Then (˜sj)j=0 q,n−1 with DS-parametrization [(dk)k=0 , (ck)k=0 ] and fn is the n−1 MHTP of degree n associated with (˜sj)j=0 . m m−1 (b) Suppose n =2m +1 and let [(ck)k=0, (dk)k=0 ] be a Hurwitz parametrization of fn. n−1 ∈K> m−1 m Then (˜sj )j=0 q,n−1 with DS-parametrization [(dk)k=0 , (ck)k=0] and fn is the n−1 MHTP of degree n associated with (˜sj)j=0 .

Proof. Denote by hn and gn the even and odd parts of fn, respectively. ∈{ } ∞ ∞ (a) Let ck := Iq and dk := Iq for all k m, m +1, ... . Then (ck)k=0 and (dk)k=0 Cq×q are sequences from > . According to [12, Proposition 8.27] there exists a sequence ∞ ∈K> ∞ ∞ (sj )j=0 q,∞ with DS-parametrization [(dk)k=0, (ck)k=0]. Using Theorem 5.4 and Deﬁnitions 2.7 and 2.10, we obtain ∞ g (−z) G (z)= 2m = − z−(j+1)s˜ σ2m,max h (−z) j 2m j=0

∈ C \ ∞ | | {| | ∈N } for all z [0, )with z >η2m, where ηn := max z : z hn . Since σ2m,max Mq ∞ 2m ≤ belongs to ≥[[0, ); (sj )j=0, ]and is concentrated on a ﬁnite set of points and hence compactly supported, we can via Lemma 3.8 conclude sj =˜sj for all j ∈ Z0,2m−1. In par- n−1 ∈K> m−1 m−1 ticular, (˜sj )j=0 q,n−1 with DS-parametrization [(dk)k=0 , (ck)k=0 ]. Proposition 7.8 n−1 yields then that fn is the MHTP of degree n associated with (˜sj )j=0 . (b) Let ck := Iq for all k ∈{m +1, m +2, ...} and let dk := Iq for all k ∈{m, m +1, ...}. ∞ ∞ Cq×q Then (ck)k=0 and (dk)k=0 are sequences from > . According to [12, Proposition 8.27] ∞ ∈K> ∞ ∞ there exists a sequence (sj)j=0 q,∞ with DS-parametrization [(dk)k=0, (ck)k=0]. Us- ing Theorem 5.4 and Deﬁnitions 2.7 and 2.10, we obtain ∞ h (−z) G (z)= 2m+1 = − z−(j+1)s˜ σ2m,min (−z)g (−z) j 2m+1 j=0

∈ C \ ∞ | | {| | ∈N } for all z [0, )with z >ρ2m+1, where ρ2m+1 := max z : z g2m+1 . Since Mq ∞ 2m+1 ≤ σ2m,min belongs to ≥[[0, ); (sj )j=0 , ]and is concentrated on a ﬁnite set of points and hence compactly supported, we can via Lemma 3.8 conclude sj =˜sj for all j ∈ Z0,2m. n−1 ∈K> m−1 m In particular, (˜sj)j=0 q,n−1 with DS-parametrization [(dk)k=0 , (ck)k=0]. Proposi- n−1 2 tion 7.8 yields then that fn is the MHTP of degree n associated with (˜sj )j=0 . Now we are able to state a useful characterization of th Hurwitz type property of monic q × q matrix polynomials in terms of the sequence of Markov parameters.

Theorem 7.10. Let n ∈{2, 3, ...} and let fn be a monic q × q matrix polynomial of n degree n with sequence of Markov parameters (˜sj)j=0. Then the following statements are equivalent:

(i) fn is a MHTP of degree n. n−1 ∈K> (ii) (˜sj )j=0 q,n−1. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 81

Proof. “(i) ⇒ (ii)”: This follows from Theorem 7.9. “(ii) ⇒ (i)”: This follows from Proposition 7.8. 2

Now we show that the correspondence between matrix Hurwitz type polynomials and ﬁnite Stieltjes positive deﬁnite sequences is bijective.

Theorem 7.11. Let n ∈{2, 3, ...} and let fn be a MHTP of degree n with sequence of n n−1 ∈K> Markov parameters (˜sj)j=0. Then there is a unique sequence (sj)j=0 q,n−1 such that n−1 n−1 n−1 fn is the MHTP of degree n associated with (sj)j=0 namely (sj)j=0 =(˜sj )j=0 .

n−1 ∈K> Proof. From Theorem 7.9 we obtain (˜sj)j=0 q,n−1 and that fn is the MHTP of degree n−1 n−1 K> n associated with (˜sj)j=0 . If (sj )j=0 is an arbitrary sequence from q,n−1 such that fn n−1 is the MHTP of degree n associated with (sj)j=0 , then Deﬁnition 6.4 and Theorem 6.1 n−1 n−1 2 yield (sj )j=0 =(˜sj)j=0 .

Now we demonstrate that a matrix Hurwitz type polynomial admits a unique Hurwitz parametrization.

Theorem 7.12. Let n ∈{2, 3, ...} and let fn be a MHTP of degree n with sequence of n ∈ N Markov parameters (˜sj)j=0. Let m be chosen such that n =2m or n =2m +1 is satisﬁed.

m−1 m−1 n−1 (a) Suppose n =2m and let [(Lk)k=0 , (Mk)k=0 ] be the DS-parametrization of (˜sj)j=0 . m−1 m−1 Then there is a unique Hurwitz parametrization [(ck)k=0 , (dk)k=0 ] of fn, namely m−1 m−1 m−1 m−1 [(ck)k=0 , (dk)k=0 ] =[(Mk)k=0 , (Lk)k=0 ]. m−1 m (b) Suppose n =2m +1 and let [(Lk)k=0 , (Mk)k=0] be the DS-parametrization of n−1 m m−1 (˜sj )j=0 . Then there is a unique Hurwitz parametrization [(ck)k=0, (dk)k=0 ] of fn, m m−1 m m−1 namely [(ck)k=0, (dk)k=0 ] =[(Mk)k=0, (Lk)k=0 ].

n−1 ∈K> Proof. From Theorem 7.9 we obtain (˜sj)j=0 q,n−1 and that fn is the MHTP of n−1 degree n associated with (˜sj )j=0 . (a) In view of Deﬁnitions 6.4 and 4.6, we can conclude from part (a) of Theorem 6.1 m−1 m−1 m−1 m−1 that [(Mk)k=0 , (Lk)k=0 ]is a Hurwitz parametrization of fn. If [(ck)k=0 , (dk)k=0 ]is an arbitrary Hurwitz parametrization of fn, then part (a) of Theorem 7.9 shows that m−1 m−1 n−1 m−1 m−1 [(dk)k=0 , (ck)k=0 ]is the DS-parametrization of (˜sj)j=0 and hence [(ck)k=0 , (dk)k=0 ] = m−1 m−1 [(Mk)k=0 , (Lk)k=0 ]. (b) In view of Deﬁnitions 6.4 and 4.6, we can conclude from part (b) of Theorem 6.1 m m−1 m m−1 that [(Mk)k=0, (Lk)k=0 ]is a Hurwitz parametrization of fn. If [(ck)k=0, (dk)k=0 ]is an arbitrary Hurwitz parametrization of fn, then part (b) of Theorem 7.9 shows that m−1 m n−1 m m−1 [(dk)k=0 , (ck)k=0]is the DS-parametrization of (˜sj)j=0 and hence [(ck)k=0, (dk)k=0 ] = m m−1 2 [(Mk)k=0, (Lk)k=0 ]. 82 A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84

8. On asymptotic stability of ODEs

As an application of the obtained results, we consider in this section the well-known Lyapunov’s stability criterion [24] for ordinary differential equations with constant coefficients in terms of orthogonal polynomials on [0, ∞)and their second kind polynomials. Let n ∈{2, 3, ...} and consider the differential equation

x˙ = Ax, (8.1) where x is an n × 1 vector, and A is an n × n matrix whose entries are real constants.

Remark 8.1. Let RA: C → C deﬁned by RA(z) := det(zIn − A)be the characteristic polynomial of the matrix A, which is a monic polynomial of degree n. Recall that the system (8.1) is asymptotically stable in the sense of Lyapunov if and only if all the zeros of the characteristic polynomial RA have a negative real part, i.e., if and only if RA is a Hurwitz polynomial.

Theorem 8.2. Let n ∈{2, 3, ...} and let A be a real n × n matrix with characteristic n polynomial RA. Denote by (˜sj )j=0 the sequence of Markov parameters of RA. Then, the n−1 system (8.1) is asymptotically stable if and only if (˜sj)j=0 is Stieltjes positive deﬁnite. M1 ∞ n−1 ≤ In this case, the set ≥[[0, ); (˜sj)j=0 , ] is non-empty and RA is the matrix Hurwitz n−1 type polynomial of degree n associated with (˜sj)j=0 .

Proof. This follows, in view of Remark 8.1, from Theorem 7.10, Remark 3.6, and Propo- sition 7.8. 2

n−1 Suppose the system (8.1) is asymptotically stable. Then, (˜sj)j=0 is Stieltjes positive ∞ ∈ definite by Theorem 8.2. In view of Proposition 3.4, there exists a sequence (sj)j=0 K> n−1 n−1 ∞ ∞ ∞ ∞ 1,∞ such that (˜sj)j=0 =(sj )j=0 . Denote by [(p1,j)j=0, (q1,j)j=0, (p2,j)j=0, (q2,j )j=0]the ∞ ∈ N SQSLOMP associated with (sj )j=0 (see Definition 5.1) and let m be chosen such that n =2m or n =2m +1is satisfied. From Theorem 8.2 and Definition 6.4 we then can conclude that m 2 m 2 (−1) p1,m(−z ) − z(−1) q1,m(−z ), if n =2m R (z)= (8.2) A m 2 m 2 (−1) q2,m(−z )+z(−1) p2,m(−z ), if n =2m +1

∈ C ∞ ∞ ∞ ∞ holds for all z . Observe that the SQSLOMP [(p1,j)j=0, (q1,j)j=0, (p2,j)j=0, (q2,j)j=0] ∞ associated with (sj )j=0 can be interpreted in terms of orthogonal polynomials with re- ∞ spect to (sj )j=0 and other related polynomials (see [4, Proposition 4.2]). Note that these polynomials then also have the corresponding orthogonality properties with respect to ∈Mq ∞ ∞ each σ ≥[[0, ); (sj )j=0, =]. We refer to [4, Section 4, Appendices D and E] for a detailed treatment of this topic. A.E. Choque Rivero / Linear Algebra and its Applications 476 (2015) 56–84 83

9. An example

In this section we consider the scalar case q =1. For j ∈ N0 let sj := j!. Then ∞ (sj )j=0 is the sequence of power moments of the exponential distribution with parameter 1. This is the measure on B[0,∞) which is given via the density h: [0, ∞) → ∞ {− } ∞ ∈K> [0, ) deﬁned by h(u) := exp u . Thus, we have (sj)j=0 1,∞. Denote by ∞ ∞ ∞ ∞ ∞ [(P1,j)j=0, (Q1,j)j=0, (P2,j)j=0, (Q2,j )j=0]the SQSLOMP associated with (sj)j=0. (Ob- ∞ serve that (P1,j)j=0 is the sequence of classical Laguerre polynomials.) Then the ﬁrst polynomials of the sequence are given by

P1,0(z)=1,P2,0(z)=1,

P1,1(z)=z − 1,P2,1(z)=z − 2, 2 2 P1,2(z)=z − 4z +2,P2,2(z)=z − 6z +6, 3 2 3 2 P1,3(z)=z − 9z +18z − 6,P2,3(z)=z − 12z +36z − 24,

Q1,0(z)=0,Q2,0(z)=1,

Q1,1(z)=1,Q2,1(z)=z − 1, 2 Q1,2(z)=z − 3,Q2,2(z)=z − 5z +2, 2 2 3 2 Q1,3(z)=z − 8z +11,Q2,3(z)=z − 11z +26z − 6.

For n ∈{2, 3, ...} we denote by fn the Hurwitz type polynomial of degree n associated ∞ with (sj )j=0. Then the ﬁrst seven polynomials are given by

f1(z)=z +1, 2 5 4 3 2 f2(z)=z + z +1, f5(z)=z + z +6z +5z +6z +2, 3 2 6 5 4 3 2 f3(z)=z + z +2z +1, f6(z)=z + z +9z +8z +18z +11z +6, 4 3 2 7 6 5 4 3 2 f4(z)=z + z +4z +3z +2, f7(z)=z + z +12z +11z +36z +26z +24z +6.

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