Helder Alexandre Sa da Costa Lima

@ua.pt

Postdoctoral researcher at the Center for Research and Development in Mathematics and Applications (CIDMA)
Universidade de Aveiro

Helder Alexandre Sa da Costa Lima


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https://researchid.co/helder_lima

RESEARCH, TEACHING, or OTHER INTERESTS

Mathematics
6

Scopus Publications

Scopus Publications

  • Bidiagonal matrix factorisations related to multiple orthogonal polynomials
    Amílcar Branquinho, Juan E.F. Díaz, Ana Foulquié-Moreno, Hélder Lima, Manuel Mañas
    Journal of Approximation Theory, 2026
  • Bidiagonal matrix factorisations associated with symmetric multiple orthogonal polynomials and lattice paths
    Hélder Lima
    Numerical Algorithms, 2025
    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 $$(r+1)$$ ( 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 $$(r+1)$$ ( 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 $$(r+1)$$ ( 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.
  • Multiple orthogonal polynomials associated with branched continued fractions for ratios of hypergeometric series
    Hélder Lima
    Advances in Applied Mathematics, 2023
  • Multiple orthogonal polynomials with respect to Gauss' hypergeometric function
    Hélder Lima, Ana Loureiro
    Studies in Applied Mathematics, 2022
    A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0,1) is studied. This type of polynomials has direct applications in the investigation of singular values of products of Ginibre random matrices and are connected with branched continued fractions and total‐positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearson‐type differential equation. The focus is on the polynomials whose indices lie on the step‐line, for which it is shown that the differentiation gives a shift in the parameters, therefore satisfying Hahn's property. We obtain Rodrigues‐type formulas for type I polynomials and functions, while a more detailed characterization is given for the type II polynomials (aka 2‐orthogonal polynomials) that include an explicit expression as a terminating hypergeometric series, a third‐order differential equation, and a third‐order recurrence relation. The asymptotic behavior of their recurrence coefficients mimics those of Jacobi–Piñeiro polynomials, based on which their asymptotic zero distribution and a Mehler–Heine asymptotic formula near the origin are given. Particular choices of the parameters and confluence relations give some known systems such as special cases of the Jacobi–Piñeiro polynomials, Jacobi‐type 2‐orthogonal polynomials, components of the cubic decomposition of threefold symmetric Hahn‐classical polynomials, and multiple orthogonal polynomials with respect to confluent hypergeometric functions of the second kind.
  • Multiple orthogonal polynomials associated with confluent hypergeometric functions
    Hélder Lima, Ana Loureiro
    Journal of Approximation Theory, 2020
  • On Müntz-type formulas related to the Riemann zeta function
    Hélder Lima
    Journal of Mathematical Analysis and Applications, 2018