Some trace formulae involving the split sequences of a Leonard pair
type: Item
instance of: scholarly article
title: Some trace formulae involving the split sequences of a Leonard pair
zbMATH Open document ID: 1085.05070
published in: Linear Algebra and its Applications
publication date: 2006-03-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0508407
review text: This paper studies so called Leonard pairs and their split sequences. Given a field \(K\) and a vector space \(V\) of finite positive dimension over \(K\), a Leonard pair is a pair \(A,\,A^{*}\) of linear transformations from \(V\) to itself that satisfies (i) there exists a basis \(\alpha\) for \(V\) such that \(\text{mat}_{\alpha,\alpha} A\) is irreducible tridiagonal and \(\text{mat}_{\alpha,\alpha}A^{*} = \text{diag}(\theta_0^{*},\theta_1^{*}, \dots,\theta_d^{*})\), (ii) there exists a basis \(\beta\) for \(V\) such that \(\text{mat}_{\alpha,\alpha} A^{*}\) is irreducible tridiagonal and \(\text{mat}_{\alpha,\alpha}A = \text{diag}(\theta_0,\theta_1, \dots, \theta_d)\). (A tridiagonal matrix is irreducible if each entry on the sub- and super-diagonal is non-zero.) The conditions (i), (ii) imply the existence of two bases \(\{u_i\}_0^d,\, \{v_i\}_0^d\) and scalars \(\{\psi_i\}_1^d, \,\{\phi_i\}_1^d\) such that \[ \begin{alignedat}{2} &Au_i= \theta_iu_i+u_{i+1}\quad (0\leq i\leq d-1),&&\qquad Au_d=\theta_du_d, \\ &A^{*}u_i= \varphi_iu_{i-1}+\theta_i^{*}u_i \quad (1\leq i\leq d),&&\qquad A^{*}u_0= \theta_0^{*}u_0,\\ &Av_i=\theta_{d-i}v_i + v_{i+1}\quad (0\leq i\leq d-1),&&\qquad Av_d= \theta_0v_d,\\ &A^{*}v_i=\phi_iv_{i-1}+\theta_i^{*}v_i \quad (1\leq i\leq d), &&\qquad A^{*}v_0 = \theta_0^{*}v_0. \end{alignedat} \] The sequence \(\{\varphi_i\}_1^d\) is called the first split sequence of the Leonard pair. For the background on Leonard pairs the reader is referred to 24 of the 25 references in the paper. After the introduction of a Leonard system and its parameter array (split sequences and certain sets of eigenvalues), formulae expressing the trace of certain polynomials in \(A,\,A^{*}\) in terms of the parameter sequences and, conversely, formulae expressing the parameter sequences in terms of quotients of traces of the polynomials are given. Finally formulae are exhibited that relate the first and second split sequences.
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D45
zbMATH DE Number: 5012998
zbMATH Keywords: tridiagonal pair
zbMATH Keywords: \(q\)-Racah polynomial
zbMATH Keywords: hypergeometric series
author: Kazumasa Nomura
author: Paul Terwilliger
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
OpenAlex ID: W2002696512
arXiv ID: math/0508407
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DOI: 10.1016/J.LAA.2005.08.019
description: scientific article