Integral polynomial sequences arising from matrix powers of order 2
type: Item
instance of: scholarly article
title: Integral polynomial sequences arising from matrix powers of order 2
zbMATH Open document ID: 1261.15038
author: Gi-Sang Cheon
author: Yongdo Lim
published in: Linear Algebra and its Applications
publication date: 2013-01-21T00:00:00Z
review text: For the matrix \( A= \left( \begin{matrix} 1 & 1 \\ x & -x \\ \end{matrix} \right) \), let \[ u_n(x)=\sum_{k=0}^{n-1}\alpha(n,k)x^k \quad\text{and}\quad v_n(x)=\sum_{k=0}^{n-1}\beta(n,k)x^k \] be the polynomials of degree \(n-1\) arising from the \(n\)-th power matrix \( A^n= \left( \begin{matrix} u_n(x) & v_n(x) \\ \ast & \ast \\ \end{matrix} \right) \). Both sequences \(\{u_n(x)\}_{n=1}^\infty\) and \(\{v_n(x)\}_{n=1}^\infty\) satisfy the recurrence relation \[ \chi_{n+2}(x)+(x-1)\chi_{n+1}(x)-2x\chi_{n}(x)=0\, . \] The authors analyze the double integral sequences \(\{\alpha(n,k)\}\) and \(\{\beta(n,k)\}\) and show that \(\alpha(n,k)=\beta(n+1,k)+\beta(n,k-1)\). Several interesting relations with, for example, a weighted Delannoy number, Vandermonde matrix, mirrored \(\Gamma\)-matrix, Riordan array, Jacobi polynomial are established.
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 6127341
zbMATH Keywords: matrix power
zbMATH Keywords: palindromic polynomial
zbMATH Keywords: weighted Delannoy number
zbMATH Keywords: mirrored \(\Gamma\)-matrix
zbMATH Keywords: recursive system
zbMATH Keywords: hypergeometric function
zbMATH Keywords: Jacobi polynomial
zbMATH Keywords: recurrence relation
zbMATH Keywords: Vandermonde matrix
zbMATH Keywords: Riordan array
reviewed by: C. M.da Fonseca
MaRDI profile type: Publication
full work available at URL: j.laa.2012.07.035
OpenAlex ID: W2064371883
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DOI: 10.1016/J.LAA.2012.07.035
description: scientific article; zbMATH DE number 6127341