Pascal-like determinants are recursive
type: Item
instance of: scholarly article
title: Pascal-like determinants are recursive
zbMATH Open document ID: 1059.05009
author: Helena Zakrajšek
author: Marko Petkovšek
published in: Advances in Applied Mathematics
publication date: 2005-01-26T00:00:00Z
review text: Let \(P=[p_{i,j}]_{i,j\geq 0}\) be a Pascal-like triangle; i.e., an infinite matrix whose entries satisfy \(p_{i,j}=\lambda p_{i-1,j}+\mu p_{i,j-1}+\nu p_{i-1,j-1}\) for all \(i,j\geq 1\), and whose first column and row satisfy linear recurrences with constant coefficients of orders \(\rho\) and \(\sigma\), respectively. Let \(d_n\) be the \(n\)th principal minor, that is the determinant of the submatrix \([p_{i,j}]_{0\leq i,j\leq n}\). In this nice paper, the authors show that the sequence \((d_n)_{n\geq 0}\) satisfies a recurrence of the type \(d_{n}=\sum_{j=1}^{\delta} c_j \omega^{jn} d_{n-j}\), where \(c_j\) are constants, \(\omega=\lambda\mu +\nu\) and \(\delta={\rho+\sigma-2 \choose \rho-1}\). When \(\lambda=\mu=1\) and \(\nu=0\), then \(\omega=1\), therefore \((d_n)_{n\geq 0}\) satisfies a linear recurrence with constant coefficients of degree \(\delta\), which confirms a conjecture of \textit{R. Bacher} [J. Théor. Nombres Bordx. 14, 19--41 (2002; Zbl 1023.11011)]. The proof proceeds in two main steps. First it is shown that \(P\) can be transformed by minor-preserving transormations into a band-diagonal form, where a matrix \(P'=[p'_{i,j}]_{i,j\geq 0}\) is called \((r,s)\)-banded if \(p'_{i,j}= 0\) unless \(-s\leq i-j\leq r\), where \(r\) and \(s\) are fixed nonnegative integers. For the authors case, they show that one can take \(r=\rho-1\) and \(s=\sigma-1\) (Corollary 3). Secondly, they prove that if \(A\) is any \((r,s)\)-banded infinite matrix, then its principal minors satisfy a linear recurrence, with nonconstant coefficients, in general, of order \({r+s\choose r}\) (Theorem 2). The proof concludes by showing that for their case the coefficients of the above recurrence are geometrical progressions whose ratios are appropriate powers of \(\omega\) (Theorem 3). \noindent The paper is written in the language of generating functions and contains several numerical examples.
reviewed by: Florian Luca
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 2131123
zbMATH Keywords: Pascal triangle
zbMATH Keywords: determinants
zbMATH Keywords: linear recurrences
describes a project that uses: DODGSON
MaRDI profile type: Publication
full work available at URL: j.aam.2003.09.004
OpenAlex ID: W2049109279
cites work: Determinants through the looking glass
cites work: Determinants of matrices related to the Pascal triangle
cites work: Linear recurrences with constant coefficients: The multivariate case
cites work: The diagonal of a double power series
cites work: Advanced determinant calculus: a complement
cites work: Evaluations of some determinants of matrices related to the Pascal triangle
cites work: Q4236280
cites work: The method of creative telescoping
cites work: Reverend Charles to the Aid of Major Percy and Fields Medalist Enrico
DOI: 10.1016/J.AAM.2003.09.004
description: scientific article; zbMATH DE number 2131123