Some discrete multiple orthogonal polynomials
type: Item
instance of: scholarly article
title: Some discrete multiple orthogonal polynomials
zbMATH Open document ID: 1021.33006
DOI: 10.1016/S0377-0427(02)00597-6
author: Walter Van Assche
author: Jorge Arvesú
author: Jonathan Coussement
published in: Journal of Computational and Applied Mathematics
publication date: 2003-05-22T00:00:00Z
review text: This very interesting paper extends a section of the Askey-tableau to the situation of discrete multiple orthogonal polynomials. The attention is restricted to so called type II systems. Given \(r\) positive measures \(d\mu_j(x)\), with support on the real line, and a multi-index \({\vec n}=(n_1,n_2,\ldots,n_r)\) of natural numbers, the multiple orthogonal polynomial \(P_{\vec n}\) associated with this multi-index is a polynomial of degree at most \(n_1+n_2+\cdots n_r\) that satisfies \[ \int_{\Delta_j} P_{\vec n}(x)x^k d\mu_j(x)=0, \quad k=0,1,\dots,n_j-1,\;j=1,2,\dots,r, \] where \(\Delta_j\) is the smallest real interval containing the support of \(d\mu_j\). Their results cover the following cases (with limit relations between them): (a) Multiple Charlier: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {a_j^k \over k!}\delta(x-k) dx\), \(a_j>0\) different; (b) Multiple Meixner I: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {(\beta)_k c_j^k \over k!}\delta(x-k)) dx\), \(0<c_j<1\) different, \(\beta>0\); (c) Multiple Meixner I: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {(\beta_j)_k c^k \over k!}\delta(x-k) dx\), \(\beta_j>0\) different, \(0<c<1\); (d) Multiple Kravchuk: \(d\mu_j(x)=(\sum_{k=0}^N {N\choose k}p_j^k(1-p_j^{n_k} \delta(x-k)) dx\), \(0<p_j<1\) different; (e) Multiple Hahn: \(d\mu_j(x)= (\sum_{k=0}^N {(\alpha_1+1)_k \over k!} {(\beta+1)_{N-k}\over (N-k)!}\delta(x-k)) dx\), \(\alpha_j>-1\) different, \(\beta>-1\), where \((s)_0=1\), \((s)_k=s(s+1)\cdots (s+k-1) (k\geq 1)\) and \(1\leq j\leq r\). For \(r=2\) several explicit formulae are given.
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A28
zbMATH DE Number: 1915218
zbMATH Keywords: multiple orthogonal polynomials
zbMATH Keywords: discrete orthogonality
zbMATH Keywords: Charlier polynomial
zbMATH Keywords: Meixner polynomial
zbMATH Keywords: Kravchuk polynomial
zbMATH Keywords: Hahn polynomial
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915218