arXiv subjects

Hélder Lima

Publications and source records attributed to Hélder Lima.

6 recordsLinked to original sources

arXiv2025

Bidiagonal matrix factorisations related to multiple orthogonal polynomials

We provide necessary and sufficient conditions for the Hessenberg recurrence matrix associated with a system of multiple orthogonal polynomials to admit a factorisation as a product of bidiagonal matrices. Using the Gauss-Borel factorisation of the moment matrix, we show that the nontrivial entries of those bidiagonal matrices can be expressed in terms of coefficients of type I or type II multiple orthogonal polynomials on the step-line with respect to the original system and its Christoffel transformations. Using the connection of multiple orthogonal polynomials with branched continued fractions, we show that the nontrivial entries of the bidiagonal matrices in the factorisation of the Hessenberg recurrence matrix correspond to the coefficients of a branched continued fraction associated with the given system of multiple orthogonal polynomials. As a case study, we present an explicit bidiagonal factorisation for the Hessenberg recurrence matrices of the Jacobi-Piñeiro polynomials and, as a limiting case, the multiple Laguerre polynomials of first kind.

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arXiv2025

Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths

The central object of study in this paper are infinite banded Hessenberg matrices admitting factorisations as products of bidiagonal matrices. In the two main novel results of this paper, we show that these Hessenberg matrices are associated with the decomposition of $(r+1)$-fold symmetric $r$-orthogonal polynomials and are the production matrices of the generating polynomials of $r$-Dyck paths. We combine the aforementioned bidiagonal matrix factorisations and the recently found connection of multiple orthogonal polynomials with lattice paths and branched continued fractions to study $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set of the complex plane and their decomposition via multiple orthogonal polynomials on the positive real line. As an explicit example, we give formulas as terminating hypergeometric series for the Appell sequences of $(r+1)$-fold symmetric $r$-orthogonal polynomials on a star-like set and show that the densities of their orthogonality measures can be expressed via Meijer G-functions on the positive real line.

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arXiv2024

Bidiagonal factorization of recurrence banded matrices in mixed multiple orthogonality

This paper demonstrates how to explicitly construct a bidiagonal factorization of the banded recurrence matrix that appears in mixed multiple orthogonality on the step-line in terms of the coeffcients of the mixed multiple orthogonal polynomials. The construction is based on the $LU$ factorization of the moment matrix and Christoffel transformations applied to the matrix of measures and the associated mixed multiple orthogonal polynomials.

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arXiv2023

Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series

The main objects of the investigation presented in this paper are branched-continued-fraction representations of ratios of contiguous hypergeometric series and type II multiple orthogonal polynomials on the step-line with respect to linear functionals or measures whose moments are ratios of products of Pochhammer symbols. This is an interesting case study of the recently found connection between multiple orthogonal polynomials and branched continued fractions that gives a clear example of how this connection leads to considerable advances on both topics. We obtain new results about generating polynomials of lattice paths and total positivity of matrices and give new contributions to the general theory of the connection between multiple orthogonal polynomials and branched continued fractions. We construct new branched continued fractions for ratios of contiguous hypergeometric series. We give conditions for positivity of the coefficients of these branched continued fractions and we show that the ratios of products of Pochhammer symbols are generating polynomials of lattice paths for a special case of the same branched continued fractions. We introduce a family of type II multiple orthogonal polynomials on the step-line associated with those branched continued fractions. We present a formula as terminating hypergeometric series for these polynomials, we study their differential properties, and we explicitly find their recurrence relation coefficients. Finally, we focus the analysis of the multiple orthogonal polynomials to the cases where the corresponding branched-continued-fraction coefficients are all positive. In those cases, the orthogonality conditions can be written using measures on the positive real line involving Meijer G-functions and we obtain results about the location of the zeros and the asymptotic behaviour of the polynomials.

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arXiv2020

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues formula. We characterise the type II polynomials via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the type II polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

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arXiv2017

On Müntz-type formulas related to the Riemann zeta function

The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical Müntz formula [4]. These formulas are derived in the critical strip and in the half-plane $Re(s)<0$. As particular cases, integral representations for products of the gamma and zeta functions are exhibited.

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