arXiv subjects

Ana Foulquié-Moreno

Publications and source records attributed to Ana Foulquié-Moreno.

27 records · Page 2Linked to original sources

arXiv2022

Hahn multiple orthogonal polynomials of type I: Hypergeometrical expressions

Explicit expressions for the Hahn multiple polynomials of type I, in terms of Kampé de Fériet hypergeometric series, are given. Orthogonal and biorthogonal relations are proven. Then, part of the Askey scheme for multiple orthogonal polynomials type I is completed. In particular, explicit expressions in terms of generalized hypergeometric series and Kampé de Fériet hypergeometric series, are given for the multiple orthogonal polynomials of type I for the Jacobi-Piñeiro, Meixner I, Meixner II, Kravchuk, Laguerre I, Laguerre II and Charlier families.

math.CA↗

arXiv2022

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices leads to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue-Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

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arXiv2022

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks

A spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization is found. The large knowledge on the spectral and factorization properties of oscillatory matrices leads to this spectral Favard theorem in terms of sequences of multiple orthogonal polynomials of types I and II with respect to a set of positive Lebesgue-Stieltjes~measures. Also a multiple Gauss quadrature is proven and corresponding degrees of precision are found. This spectral Favard theorem is applied to Markov chains with $(p+2)$-diagonal transition matrices, i.e. beyond birth and death, that admit a positive stochastic bidiagonal factorization. In the finite case, the Karlin-McGregor spectral representation is given. It is shown that the Markov chains are recurrent and explicit expressions in terms of the orthogonal polynomials for the stationary distributions are given. Similar results are obtained for the countable infinite Markov chain. Now the Markov chain is not necessarily recurrent, and it is characterized in terms of the first measure. Ergodicity of the Markov chain is discussed in terms of the existence of a mass at $1$, which is an eigenvalue corresponding to the right and left eigenvectors.

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arXiv2022

Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

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-Piñeiro 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.

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arXiv2022

Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. In this paper conditions, in terms of continued fractions, for an oscillatory tetradiagonal Hessenberg matrix to have such positive bidiagonal factorization are found. Oscillatory tetradiagonal Toeplitz matrices are taken as a case study of matrix that admits a positive bidiagonal factorization. Moreover, it is proved that oscillatory banded Hessenberg matrices are organized in rays, with the origin of the ray not having the positive bidiagonal factorization and all the interior points of the ray having such positive bidiagonal factorization.

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arXiv2022

Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

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-Piñeiro 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-Piñeiro 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.

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arXiv2022

Matrix Jacobi Biorthogonal Polynomials via Riemann-Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-PIV equations are obtained for the three term recurrence relation coefficients.

math.CA↗

arXiv2021

Hypergeometric Multiple Orthogonal Polynomials and Random Walks

The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christoffel-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large $n$ limit. These random walks are beyond birth and death, as they describe a chain in where transitions to the two previous states are allowed, or in the dual to the two next states.The corresponding Karlin-McGregor representation formula is given for these new Markov chains. The regions of hypergeometric parameters where the Markov chains are recurrent or transient are given. Stochastic factorizations, in terms of pure birth and of pure death factors, for the corresponding Markov matrices of types I and II, are provided.Twelve uniform Jacobi matrices and the corresponding random walks, related to a Jacobi matrix of Toeplitz type, and theirs stochastic or semi-stochastic matrices (with sinks and sources), that describe Markov chains beyond birth and death, are found and studied. One of these uniform stochastic cases, which is a recurrent random walk, is the only hypergeometric multiple random walk having a uniform stochastic factorization. The corresponding weights, Jacobi and Markov transition matrices and sequences of type II multiple orthogonal polynomials are provided. Chain of Christoffel transformations connecting the stochastic uniform tuples between them, and the semi-stochastic uniform tuples, between them, are presented.

math.CA↗

arXiv2021

Multiple Orthogonal Polynomials and Random Walks

Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding Markov chains (or 1D random walks) allow, in one transition, to reach for the N-th previous states, to remain in the state or reach for the immediately next state. The dual Markov chains allow, in one transition, to reach for the N-th next states, to remain in the state or reach for immediately previous state. The connection between both dual Markov chains is discussed at the light of the Poincaré's theorem on ratio asymptotics for homogeneous linear recurrence relations and the Christoffel-Darboux formula within the sequence of multiple orthogonal polynomials and linear forms of type I. The Karlin-McGregor representation formula is extended to both dual random walks, and applied to the discussion of the corresponding generating functions and first-passage distributions. Recurrent or transient character of the Markov chain is discussed. Steady state and some conjectures on its existence and the relation with mass points are also given. The Jacobi-Piñeiro multiple orthogonal polynomials are taken as a case study of the described results. For the first time in the literature, an explicit formula for the type~I Jacobi--Piñeiro polynomials is determined. The region of parameters where the Markov chains are recurrent or transient is given, and it is conjectured that when recurrent, the Markov chains are null recurrent and, consequently, the expected return times are infinity. Examples of recurrent and transient Jacobi--Piñeiro random walks are constructed explicitly.

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