Identities among certain triangular matrices
type: Item
instance of: scholarly article
title: Identities among certain triangular matrices
zbMATH Open document ID: 0598.15009
DOI: 10.1016/0024-3795(86)90142-4
author: P. R. Vein
published in: Linear Algebra and its Applications
publication date: 1986-01-01T00:00:00Z
review text: From author's summary: A family of simple matrices \(Q_ m\) is defined and a number of relations among its members are given. Two sets of triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 3964057
zbMATH Keywords: matrix identities
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: triangular matrices
zbMATH Keywords: Laguerre
zbMATH Keywords: Hermite
zbMATH Keywords: Bernoulli
zbMATH Keywords: Euler
zbMATH Keywords: Bessel
zbMATH Keywords: Stirling numbers
zbMATH Keywords: Eulerian numbers
zbMATH Keywords: similarity transformations
zbMATH Keywords: inverse scalar relations
zbMATH Keywords: umbral calculus
MaRDI profile type: Publication
full work available at URL: 0024-3795(86)90142-4
OpenAlex ID: W1967641284
cites work: Matrices which generate families of polynomials and associated infinite series
cites work: Q3661894
cites work: Q5820721
cites work: Generating Functions for Bessel and Related Polynomials
cites work: Q4769056
cites work: Q5589310
cites work: Q5535148
cites work: On representations of the braid group
cites work: A Generalization of the Geometric Series
cites work: On a Generlization of the Geometric Series
cites work: Q3884111
cites work: Generalized Eulerian numbers
cites work: Persymmetric determinants 5. Families of overlapping coaxial equivalent determinants
cites work: Induced Transformations of the Derivative-Vector
cites work: Some Finite Inversion Formulae
cites work: Some inversion formulas
cites work: Q5589311
cites work: Q3658304
cites work: Q4093447
cites work: The umbral calculus
cites work: On the foundations of combinatorial theory. VIII: Finite operator calculus
cites work: Die Potenzreihe \(\sum n^ m z^ n\), \(m \in N\)
cites work: Polynomial Translation Groups
description: scientific article; zbMATH DE number 3964057