arXiv subjects

Juan E. F. Díaz

Publications and source records attributed to Juan E. F. Díaz.

6 recordsLinked to original sources

arXiv2026

Piñeiro mixed orthogonal polynomials and bidiagonal factorization

We study mixed multiple orthogonal polynomials associated with rank-one matrices of measures and their recurrence relations. On the step line we identify the corresponding banded recurrence matrix through the Gauss--Borel factorization of the moment matrix. Elementary Christoffel transformations lead to a factorization of this recurrence matrix into lower and upper bidiagonal factors. For the mixed Piñeiro system we derive contour-integral and generalized hypergeometric representations and formulas for the recurrence and factorization coefficients.

math.CA↗

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.

math.CA↗

arXiv2024

Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion coefficients of Hahn multiple orthogonal polynomials are derived. By leveraging the multiple Askey scheme and the recently discovered explicit hypergeometric representation of type I multiple Hahn polynomials, corresponding explicit hypergeometric representations are provided for the type I polynomials and recursion coefficients of all the aforementioned descendants within the Askey scheme. Additionally, integral representations for these families within the Hahn class in the Askey scheme are presented. The multiple Askey scheme is further completed by providing the corresponding limits for the weights, polynomials, and recurrence coefficients.

math.CA↗

arXiv2023

Bidiagonal factorization of the recurrence matrix for the Hahn multiple orthogonal polynomials

This paper explores a factorization using bidiagonal matrices of the recurrence matrix of Hahn multiple orthogonal polynomials. The factorization is expressed in terms of ratios involving the generalized hypergeometric function ${}_3F_2$ and is proven using recently discovered contiguous relations. Moreover, employing the multiple Askey scheme, a bidiagonal factorization is derived for the Hahn descendants, including Jacobi-Piñeiro, multiple Meixner (kinds I and II), multiple Laguerre (kinds I and II), multiple Kravchuk, and multiple Charlier, all represented in terms of hypergeometric functions. For the cases of multiple Hahn, Jacobi-Piñeiro, Meixner of kind II, and Laguerre of kind I, where there exists a region where the recurrence matrix is nonnegative, subregions are identified where the bidiagonal factorization becomes a positive bidiagonal factorization.

math.CA↗

arXiv2023

Markov Chains and Multiple Orthogonality

In this work we survey on connections of Markov chains and the theory of multiple orthogonality. Here we mainly concentrate on give a procedure to generate stochastic tetra diagonal Hessenberg matrices, coming from some specific families of multiple orthogonal, such as the ones of Jacobi--Piñeiro and Hypergeometric Lima--Loureiro. We show that associated with a positive tetra diagonal nonnegative bounded Hessenberg matrix we can construct two stochastic tetra diagonal ones. These two stochastic tridiagonal nonnegative Hessenberg matrices are shown to be, enlightened by the Poincaré theorem, limit transpose of each other.

math.PR↗

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↗