Multiple orthogonal polynomials associated with confluent hypergeometric functions
type: Item
instance of: scholarly article
title: Multiple orthogonal polynomials associated with confluent hypergeometric functions
zbMATH Open document ID: 1453.33006
author: Hélder Lima
author: Ana F. Loureiro
published in: Journal of Approximation Theory
publication date: 2020-10-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/2001.06820
review text: Given a pair of measures \( (\mu_{0}, \mu_{1})\) supported on the real line, type I multiple orthogonal polynomials for \(\vec{n}=(n_{0}, n_{1})\) are given by a vector \(( A^{(0)}_{\vec{n}}(x), A^{(1)}_{\vec{n}}(x))\) of polynomials with deg \(A^{(j)}_{\vec{n}}(x) \leq n_{j} -1, j=0,1\) satisfying the orthogonality and normalisation conditions \(\int x^j A^{(0)}_{\vec{n}}(x) d\mu_{0} + \int x^j A^{(1)}_{\vec{n}}(x) d\mu_{1}=0,\) if \(0\leq k\leq |\vec{n}| -2,\) and \(=1,\) if \(k= |\vec{n}| -1 . \) On the other hand, type II multiple orthogonal polynomials for \(\vec{n}=(n_{0}, n_{1})\) are monic polynomials \(P_{\vec{n}}(x)\) of degree \(|\vec{n}|\) which satisfy the orthogonality conditions \(\int x^k P_{\vec{n}}(x) d\mu_{j}=0, j=0, 1, k= 0,1, \cdots, n_{j}-1.\) The above orthogonality conditions give a non-homogeneous system of \(|\vec{n}|\) linear equations for the unknown coefficients of the polynomials \( A^{(0)}_{\vec{n}}(x), A^{(1)}_{\vec{n}}(x)\) and \(P_{\vec{n}}(x),\) respectively. For more information about multiple orthogonal polynomials, see [\textit{E. M. Nikishin} and \textit{V. N. Sorokin}, Rational approximations and orthogonality. Transl. from the Russ. by Ralph P. Boas. Transl. ed. by S. Gelfand. Providence, RI: American Mathematical Society (1991; Zbl 0733.41001)], as well as Chapter 23 in [\textit{M. E. H. Ismail}, Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. Cambridge: Cambridge University Press (2005; Zbl 1082.42016)]. In the paper under review the authors study multiple orthogonal polynomials with respect to two absolutely continuous measures supported on the positive real line and having an integral representation in terms of weight functions \(\mathcal{W}(x;a,b;c)\) and \(\mathcal{W}(x;a, b;c+1),\) where \(\mathcal{W}(x;a,b;c)= \frac{\Gamma(c)}{\Gamma(a) \Gamma(b)} e^{-x} x^{a-1} \textbf{U }(c-b, a-b+1;x)\) with \(a,b,c\) positive real numbers, \(c> max(a, b)\) and \(\textbf{U}( \al…
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33C15
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 7268510
zbMATH Keywords: multiple orthogonal polynomials
zbMATH Keywords: confluent hypergeometric function
zbMATH Keywords: Nikishin system
zbMATH Keywords: Rodrigues formula
zbMATH Keywords: generalised hypergeometric series
zbMATH Keywords: 2-orthogonal polynomials
zbMATH Keywords: Hahn-classical
zbMATH Keywords: 3-fold symmetric
reviewed by: Francisco Marcellán
MaRDI profile type: Publication
OpenAlex ID: W3089155537
arXiv ID: 2001.06820
cites work: The genetic sums' representation for the moments of a system of Stieltjes functions and its application
cites work: On \(d\)-symmetric classical \(d\)-orthogonal polynomials
cites work: On two-orthogonal polynomials related to the Bateman's \(J_n^{u,v}\)-function
cites work: The second Painlev equation, its hierarchy and associated special polynomials
cites work: Q5459902
cites work: LES POLYNÔMES ORTHOGONAUX „CLASSIQUES“ DE DIMENSION DEUX
cites work: Une caractérisation des polynômes \(d\)-orthogonaux ''classiques''
cites work: Q5820721
cites work: Nikishin systems are perfect
cites work: Nikishin systems are perfect. The case of unbounded and touching supports
cites work: Q3084090
cites work: Matrix Analysis
cites work: Q5703969
cites work: Special Functions, Stieltjes Transforms and Infinite Divisibility
cites work: The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials
cites work: Extensions of Karlsson–Minton summation theorem and some consequences of the first Miller–Paris transformation
cites work: Singular values of products of random matrices and polynomial ensembles
cites work: Three-fold symmetric Hahn-classical multiple orthogonal polynomials
cites work: L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux
cites work: Two-dimensional orthogonal polynomials, their associated sets and the co- recursive sets
cites work: Semi-classical character and finite-type relations between polynomial sequences
cites work: Generalized Hypergeometric Function of Unit Argument
cites work: ON SIMULTANEOUS PADÉ APPROXIMANTS
cites work: Nearest neighbor recurrence relations for multiple orthogonal polynomials
cites work: Multiple orthogonal polynomials associated with macdonald functions
cites work: Vector orthogonal relations. Vector QD-algorithm
cites work: Q5792555
DOI: 10.1016/J.JAT.2020.105484
describes a project that uses: Digital Library of Mathematical Functions
description: scientific article