Hankel determinants of a linear combination of three successive Catalan numbers
type: Item
instance of: scholarly article
title: Hankel determinants of a linear combination of three successive Catalan numbers
zbMATH Open document ID: 1275.33014
author: Belgacem Bouras
published in: Mediterranean Journal of Mathematics
publication date: 2013-05-17T00:00:00Z
review text: The author evaluates the determinants of Hankel matrices, whose elements are a linear combination of three successive shifted Catalan numbers. Let \[ a_n=\alpha C_{n+k}+\beta C_{n+k+1}+\gamma C_{n+k+2},\;n\geq0,\;\alpha,\beta,\gamma \in \mathbb{C}, \] where \(C_n=\frac{\binom{2n}{n}}{n+1}\), \(n\geq0\), be Catalan numbers and \(k\) an arbitrary positive integer. By \(h_{n,k}(\gamma,\beta,\alpha)=\det(a_{i+j})_{0\leq i,j\leq n}\), he denotes the Hankel determinant of order \(n\) of \(a_0,a_1,\dots\). If \(\gamma\neq0\), the author shows that \(h_n(\gamma,\beta,\alpha)\) can be expressed in terms of some modified monic Jacobi polynomials. The particular case when \(\gamma=0\) and \(\beta\neq0\) is deduced, and, finally, some particular cases when \(\gamma\neq0\) are studied.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 6164328
zbMATH Keywords: Hankel determinants
zbMATH Keywords: Catalan numbers
zbMATH Keywords: Jacobi polynomials
reviewed by: M. Mejri
MaRDI profile type: Publication
full work available at URL: s00009-012-0200-x
OpenAlex ID: W2006795202
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DOI: 10.1007/S00009-012-0200-X
description: scientific article; zbMATH DE number 6164328