Hankel determinants of some polynomials arising in combinatorial analysis
type: Item
instance of: scholarly article
title: Hankel determinants of some polynomials arising in combinatorial analysis
zbMATH Open document ID: 0956.05016
DOI: 10.1023/A:1019197311168
author: Jet Wimp
published in: Numerical Algorithms
publication date: 2000-11-22T00:00:00Z
review text: Evaluations of special Hankel determinants (this is a determinant of the form \(\det(c_{i+j})_{1\leq i,j\leq n}\)) have always fascinated many people, as there are a lot of beautiful such results. It is classical, but unfortunately not wide-spread enough, that the deep reason for this phenomenon is that evaluations of Hankel determinants are intimately tied to orthogonal polynomials. More precisely, given a sequence of orthogonal polynomials with an associated measure, the \(i\)th moment of which is \(c_i\), the Hankel determinant \(\det(c_{i+j})_{1\leq i,j\leq n}\) is simply a product of the coefficients which appear in the three-term recurrence that the orthogonal polynomials must satisfy, thanks to Favard's theorem. In particular, most of the found evaluations of Hankel determinants do in fact come from known facts about hypergeometric orthogonal polynomials in the Askey scheme. The purpose of the article under review is to put emphasis on this point, and to discuss this matter by considering several examples. (For a similar treatment, more examples and references, see Section~2.7 and Theorem~52, plus the subsequent discussion, in the reviewer's survey [Sémin. Lothar. Comb. 42, B42q (1999; Zbl 0923.05007)].) In particular, the author derives an evaluation of a Hankel determinant with entries being certain hypergeometric \(_2F_1\)-series, and shows that several previously found Hankel determinant evaluations in the literature follow on specializing his result. Moreover he derives several evaluations which apparently have not been recorded earlier. The author also discusses three Hankel determinant evaluations, which he is not able to prove. However, two of these do in fact appear in the literature. Namely, the first of these, the evaluation of \(\det(B_{i+j})_{0\leq i,j\leq n}\), where \(B_k\) denotes the \(k\)th Bernoulli number, appears in \textit{W. A. Al-Salam} and \textit{L. Carlitz} [Portugaliae Math. 18, 91-99 (1959; 093.01504)]. The second, the evaluation …
reviewed by: C. Krattenthaler
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1534770
zbMATH Keywords: Hankel determinants
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: derangement numbers
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: \(q\)-ultraspherical polynomials
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Bell numbers
zbMATH Keywords: Meixner polynomials
zbMATH Keywords: Meixner-Po
MaRDI profile type: Publication
full work available at URL: a:1019197311168
OpenAlex ID: W203349441
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description: scientific article; zbMATH DE number 1534770