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Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima-Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi-Pi\~neiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin., Comment: This is the third part of the splitting of the paper arXiv:2203.13578 into three. 15 pages and 1 figure

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{\mu-\frac12}$, $\mu > -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first family consists of monomial polynomials $\mathsf{V}_{\mathbf{k},n}(x,t) = t^{n-|\mathbf{k}|} x^\mathbf{k} + \cdots$ for $\mathbf{k} \in \mathbb{N}_0^d$ with $|\mathbf{k}| \le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|\mathbf{k}|} x^\mathbf{k} + \mathsf{P}$ with $\deg \mathsf{P} \le n-1$, and we will provide an explicit construction for $\mathsf{V}_{\mathbf{k},n}$. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when $w$ is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal., Comment: 24 pp

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of B¨acklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex., FCT-FCCN (b-on)

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss-Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre-Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi-Pi\~neiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi-Pi\~neiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes-Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial., Comment: Revised version, completely new section on general Christoffel and Geronimus for multiple orthogonal polynomials on the stepline

Recently the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, has been presented for matrix biorthogonal polynomials, in particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coe cients first order matrix polynomials. We will explore this discussion, present some achievements and consider some new examples of weights for matrix biorthogonal polynomials. published

We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix function. Next, from an arbitrary sequence of type II multiple orthogonal polynomials with respect to a set of d linear functionals, we obtain a total of d + 1 sequences of type II multiple orthogonal polynomials, which can be used to construct a new sequence of symmetric type II multiple orthogonal polynomials. We also prove a Favard-type result for certain sequences of matrix multiple orthogonal polynomials satisfying a matrix four-term recurrence relation with matrix coefficients. Finally, we provide an example concerning multiple Hermite polynomials.

In this paper we study matrix biorthogonal polynomials sequences that satisfy a nonsymmetric three term recurrence relation with unbounded matrix coefficients. The outer ratio asymptotics for this family of matrix biorthogonal polynomials is derived under quite general assumptions. Some illustrative examples are considered. Acknowledgements. AB acknowledges Centro de Matemática da Universidade de Coimbra (CMUC) UID/MAT/00324/2019, funded by the Portuguese Govern- ment through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. JCGA & FM thanks financial support from the Spanish “Ministerio de Economía y Competitividad” research project MTM2015-65888-C4-2-P, Ortogonalidad y aproximación; teoría y aplicaciones

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In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials on the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first degree matrix polynomials, is given. All of these are applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because of the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of a hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete PainleveI equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painleve I equation is found. (c) 2020 Elsevier Inc. All rights reserved.

The correspondences between dynamics of q-Toda and q-Volterra equations for the coefficients of the Jacobi operator and its resolvent function are established. The orthogonal polynomials associated with these Jacobi operators satisfy an Appell condition, with respect to the q-difference operator $$D_{q}$$ . Lax type theorems for the point spectrum of the Jacobi operators associated with these equations are obtained. Examples related with the big q-Legendre, discrete q-Hermite I, and little q-Laguerre orthogonal polynomials and q-Toda and q-Volterra equations are given.

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One derives discrete dynamical systems related to Laguerre–Hahn orthogonal polynomials. One studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t -dependence on the coefficients of the Riccati differential equation for the related Stieltjes function.

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.

In this work we characterize a high-order Toda lattice in terms of a family of matrix polynomials orthogonal with respect to a complex matrix measure. In order to study the solution of this dynamical system, we give explicit expressions for the Weyl function, generalized Markov function, and we also obtain, under some conditions, a representation of the vector of linear functionals associated with this system. We show that the orthogonality is embedded in these structure and governs the high-order Toda lattice. We also present a Lax-type theorem for the point spectrum of the Jacobi operator associated with a Toda-type lattice

In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced.

We prove the semi-classical character of some sequences of orthogonal polynomials, say , , related through relations of the following type: , where denotes the monic polynomial associated with the pth order derivative of , and are complex numbers. The case is studied for a pair of orthogonal polynomials whose corresponding orthogonality measures are coherent. The relation is shown to give a characterization for the semi-classical character of .

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Pade type are also discussed. Finally, a Markov's type theorem is presented.

9 pages, no figures.-- MSC1991 codes: 41A20, 41A25, 41A28.-- Issue title: "9th OPSFA Conference" (9th Conference on Orthogonal Polynomials, Special Functions and Applications, Luminy, Marseille, France, 2-6 July 2007). Zbl#: Zbl pre05624871 Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szegö-type functions. In the procedure, an approach from the Riemann–Hilbert problem plays a fundamental role. The first author’s research was supported by CMUC/FCT. The second author’s research was supported by grants MTM 2006-13000-C03-02 from Ministerio de Ciencia y Tecnología and CCG 06-UC3M/ESP-0690 of Universidad Carlos III de Madrid-Comunidad de Madrid and by grant SFRH/BPD/ 31724/2006 from Fundação para a Ciência e a Tecnologia. The third author’s research was supported by UI Matemática e Aplicações from University of Aveiro. Publicado

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Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of Backlund transformations such that each new generalized Toda solution is related to a generalized Volterra solution.

In this paper we introduce the Δ-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like ( 1 + x 2 ) 1 − t μ ( x ) where μ is a given positive Borel measure. Moreover, the Δ-Volterra lattice is related to the Δ-Toda lattice from Miura or Backlund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ and the characterization of the point spectrum of a Jacobian operator that satisfies a Δ-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Δ-Volterra and Δ-Toda lattices, and connect this example with the results presented in the paper.

In this paper a discretization of Toda equations is analyzed. The correspondence between these Δ-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. It is shown that the spectral measure of these operators evolve in t like ${(1+x)}^{1-t}\;{\rm{d}}\mu (x)$ where ${\rm{d}}\mu$ is a given positive Borel measure. The Lax pair for the Δ-Toda equations is derived and characterized in terms of linear functionals, where orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Δ appear in a natural way. In order to illustrate the results of the paper we work out two examples of Δ-Toda equations related with Jacobi and Laguerre orthogonal polynomials.

9 pages, no figures.-- MSC1991 code: Primary 42C05. MR#: MR2016675 (2004k:41025) Zbl#: Zbl 1035.41010 We improve the class of indices for which normality takes place in a Nikishin system and apply this in Hermite–Padé approximation of such systems of functions. A.B. thanks support from Grants PRAXIS XXI BCC-22201/99 and INTAS 00-272, J.B. from grant CONACYT 32181-E, A.F.M. from Grants PRAXIS XXI BPD-20396/99 and INTAS 00-272, G.L.L. from Grants PRAXIS XXI BCC-22201/99, BFM 2000-0206-C04-01 and INTAS 00-272. Publicado

A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to p > 1 weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order p + 1. We also obtain explicit formulas and recurrence relations for these polynomials.

Tridiagonal operators with complex coefficients are considered. The correspondence between dynamics of high order Toda equations for the coefficients of the operator and its resolvent function is established. It gives a method to solve an inverse problem—integration of high order Toda equations with complex initial data—based on Padé approximants and continued fractions for the resolvent function.

A characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices is stated and proved. This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.

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Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product \langle f, g \rangle = ∈t_{E} f(ξ) \overline{g(ξ)} ρ(ξ) |d ξ|+ f(Z) A g(Z)^H, where E is a rectifiable Jordan curve or arc in the complex plane

Let μ be a positive Borel measure on the unit circle with infinite support and φn(z)=φn(z,μ) be monic orthogonal polynomials with respect to μ. We give a full description of the class of measures possessing the following property: there exists a sequence of complex numbers (αn) such that monic polynomials ψn=φn − αnφn−1 are also orthogonal with respect to a certain measure v.

This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure $\mu$, i.e., \begin{equation*} \frac{1}{(x-c)}d\mu (x)+N\delta (x-c), \end{equation*} for some free parameter $N \in \mathbb{R}_{+}$ and shift $c$. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass $N$ tends to infinity as well as we characterize the precise values of $N$ such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure $\mu$. When $\mu$ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation $c$ is located outside of the support of $\mu$., Comment: Some minor typos in the text have been corrected. B. Murgante et al. (Eds.): ICCSA 2014 - Lecture Notes in Computer Science (LNCS), 8579 (Part I) (2014), 44-59. Springer International Publishing Switzerland. ISSN 0302-9743. ISBN13: 978-3-319-09143-3. ISBN10: 3-319-09143-3

Transformations of the measure of orthogonality for orthogonal polynomials, namely Freud transformations, are considered. Jacobi matrix of the recurrence coefficients of orthogonal polynomials possesses an isospectral deformation under these transformations. Dynamics of the coefficients are described by generalized Toda equations. The classical Toda lattice equations are the simplest special case of dynamics of the coefficients under the Freud transformation of the measure of orthogonality. Also dynamics of Hankel determinants, its minors and other notions corresponding to the orthogonal polynomials are studied.

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Matrix Sylvester differential equations are introduced in the study of Laguerre-Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre-Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre-Hahn orthogonal polynomials.

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(φu)=ψu. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.

The correspondence between a high-order non-symmetric difference operator with complex coefficients and t h e evolution of an operator defined by a Lax pair is established. The solution of t h e discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on t he evolutions of t he matrical moments.

High-order non-symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems.

In this work we give an interpretation of a (s (d + 1 ) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials. We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions. We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite-Pade approximation in matrix form is given.

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16 pages, no figures.-- 2000 MSC code: Primary 42C05. MR#: MR2432558 (2009e:42048) Zbl#: Zbl 1149.42013 In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if (μ0,μ1) is a coherent pair of measures on the unit circle, then μ0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with μ1 is a specific rational transformation of the linear functional corresponding to μ0. Some examples are given. The work of the first and fourth author was supported by (CMUC) Department of Mathematics, University of Coimbra. The second author would like to thank Unidade de investigação (Matemática e Aplicações) da Universidade de Aveiro. The work of the third author was supported by Dirección General de Investigación (Ministerio de Educación y Ciencia) of Spain under Grant MTM 2006-13000-C03-02. The work of M.N. Rebocho was supported by FCT, Fundação para a Ciência e Tecnologia, with Grant ref. SFRH/BD/25426/2005. Publicado

Sequences of orthogonal polynomials on the unit circle whose Caratheodory function satisfies a Riccati-type differential equation with polynomial coefficients are studied. We deduce discrete Lax equations which lead to difference equations for the corresponding sequences of reflection parameters, and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur.

Bonan et al. (1987) gave an apparent generalization of semiclassical orthogonal polynomial sequences for positive measures as an inverse problem for orthogonal polynomials. We study a more general situation for regular orthogonal polynomials. The connection between the corresponding linear functions is obtained. The basic result is the semiclassical character of such functionals.

Let u be a Hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Caratheodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for F, zAF^'=BF^2+CF+D, and a distributional equation for u, D(Au)=B@?u^2+C@?u+H@?L, where L is the Lebesgue functional, and the polynomials B@?,C@?,H@? are defined in terms of the polynomials A,B,C,D.

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We prove that one characterization for the classical orthogonal polynomials sequences (Hermite, Laguerre, Jacobi and Bessel) cannot be extended to the semi-classical ones.