Generalized Pascal functional matrix and its applications
type: Item
instance of: scholarly article
title: Generalized Pascal functional matrix and its applications
zbMATH Open document ID: 1115.15012
author: Yongzhi Yang
author: Catherine Micek
published in: Linear Algebra and its Applications
publication date: 2007-05-10T00:00:00Z
review text: Let \(f(t;x)\) be a function of \(t\) with parameter \(x\) for which the \(n\)-th order derivatives with respect to \(t\) exist. The generalized Pascal functional matrix \({\mathcal P}_n[f(t;x)]\) is an \((n+1)\times(n+1)\) matrix such that \(({\mathcal P}_n[f(t;x)])_{ij}= {\binom ij}f^{(i-j)}(t;x)\) if \(i\geq j\) for \(i,j=0,1,2,\ldots,n\), and \(0\) otherwise. The authors show, if \({\mathcal P}_n[f(t;x)]\) and \({\mathcal P}_n[g(t;x)]\) are any two \((n+1)\times (n+1)\) Pascal functional matrices, then \({\mathcal P}_n[f(t;x)]{\mathcal P}_n[g(t;x)]={\mathcal P}_n[f(t;x)g(t;x)]={\mathcal P}_n[g(t;x)]{\mathcal P}_n[f(t;x)].\) They also prove some combinatorial identities, e.g. Tepper's identity, and they determine the \(LU\) decomposition of some well-known matrices.
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 5151563
zbMATH Keywords: generalized Pascal functional matrix
zbMATH Keywords: Pascal matrices
zbMATH Keywords: combinatorial identities
zbMATH Keywords: LU decomposition
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: j.laa.2006.12.014
OpenAlex ID: W2074927252
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DOI: 10.1016/J.LAA.2006.12.014
description: scientific article; zbMATH DE number 5151563