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Amilcar Branquinho

Publications and source records attributed to Amilcar Branquinho.

7 recordsLinked to original sources
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

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

Monomial and Rodrigues orthogonal polynomials on the cone

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{μ-\frac12}$, $μ> -\frac 12$, 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 $°\mathsf{P} \le n-1$, 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.

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arXiv2019

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV

In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights ---which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann-Hilbert problem are derived. An explicit and general example is presented to illustrate the theoretical results of the work. Related matrix eigenvalue problems for second order matrix differential operators and non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

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arXiv2018

Matrix Biorthogonal Polynomials: eigenvalue problems and non-Abelian discrete Painlevé equations

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 in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first order matrix polynomials, is given. All these is 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 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 an hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete Painlevé I 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 Painlevé I equation is found.

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arXiv2017

Matrix biorthogonal polynomials, associated polynomials and functions of the second kind

In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A sort of Poincaré's theorem for the ratio of two consecutive matrix functions solutions of a linear difference equation is also presented. Some new formulas connecting these families of matrix functions are given.

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