Addition formulas for polynomials built on classical combinatorial sequences
type: Item
instance of: scholarly article
title: Addition formulas for polynomials built on classical combinatorial sequences
zbMATH Open document ID: 1025.11004
DOI: 10.1016/S0377-0427(99)00120-X
author: Christian Radoux
published in: Journal of Computational and Applied Mathematics
publication date: 2003-04-10T00:00:00Z
review text: This short article contains quite a number of interesting results concerning Catalan numbers, Catalan polynomials, Stirling numbers of the second kind, Bell polynomials, Hermite polynomials and derangement polynomials in relation to explicit computation of several Hankel determinants as a continuation of the author's earlier results (the references list five papers, all by the present author). The first result states that the Hankel matrix of any order built on Catalan numbers has determinant 1. Then he obtains addition formulae and computes the Hankel determinant for Catalan polynomials and related ones defined by \(Q_n(x)= \sum^n_{k=0} {n\choose k}^2 x^k\), as well as for Bell polynomials related to Stirling numbers of the second kind \[ \det\bigl( B_{i+j}(x) \bigr)_{i,j=0, \dots,n}= x^{n(n+1)/2} \prod^n_{k=0} k!. \] A purely combinatorial enumerative proof for this formula was recently given by \textit{R. Ehrenborg} [Am. Math. Mon. 107, 557-560 (2000; Zbl 0985.15006)]. The author's earlier proof [Bull. Soc. Math. Belg., Sér. B 31, 49-55 (1979; Zbl 0439.10006)] used a Silvester's identity.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 1423337
zbMATH Keywords: Catalan numbers
zbMATH Keywords: Catalan polynomials
zbMATH Keywords: Stirling numbers
zbMATH Keywords: Bell polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: derangement polynomials
zbMATH Keywords: Hankel determinants
MaRDI profile type: Publication
cites work: Generalized Catalan numbers
cites work: Q4298897
cites work: Q3880900
cites work: Q3977043
cites work: Hankel determinant constructed on polynomials associated with the number of derangements
Wikidata QID: Q126351595
description: scientific article; zbMATH DE number 1423337