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Here we show that Jackson integral measures for the polynomials of the big qJacobi family are the consequence of the recovery of selfadjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of selfadjointness is achieved in an extended \u00272 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrodinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the twounbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the qMeixner (qCharlier) polynomials do not form a complete basis of the \u00272 Hilbert space, based on the fact that the dual qMeixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. 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Orthogonal polynomials from Hermitian matrices. II
http://hdl.handle.net/10091/00020587
d725aac3ba84466682d3719d0cf61252
Name / File  License  Actions  

1604.00714v2.pdf (617.0 kB)

© 2018 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. / The following article appeared in JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018) and may be found at (http://doi.org/10.1063/1.5021462).

item type  学術雑誌論文 / Journal Article(1)  

公開日  20180508  
タイトル  
タイトル  Orthogonal polynomials from Hermitian matrices. II  
言語  
言語  eng  
資源タイプ  
資源  http://purl.org/coar/resource_type/c_6501  
タイプ  journal article  
著者 
Odake, Satoru
× Odake, Satoru× Sasaki, Ryu 

信州大学研究者総覧へのリンク  
氏名  Odake, Satoru  
URL  http://soarrd.shinshuu.ac.jp/profile/ja.uhLeuUkV.html  
出版者  
出版者  AMER INST PHYSICS  
引用  
内容記述タイプ  Other  
内容記述  JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018)  
書誌情報 
JOURNAL OF MATHEMATICAL PHYSICS 巻 59, 号 1, p. 013504, 発行日 201801 

抄録  
内容記述タイプ  Abstract  
内容記述  This is the second part of the project "unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of Hermitian matrices." In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional Hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big qJacobi family are the consequence of the recovery of selfadjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of selfadjointness is achieved in an extended '2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrodinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the twounbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the qMeixner (qCharlier) polynomials do not form a complete basis of the '2 Hilbert space, based on the fact that the dual qMeixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the qMeixner polynomials is obtained by constructing the duals of the dual qMeixner polynomials which require the twocomponent Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big qJacobi family and their duals and qMeixner (qCharlier) polynomials. Published by AIP Publishing.  
資源タイプ（コンテンツの種類）  
内容記述タイプ  Other  
内容記述  Article  
ISSN  
収録物識別子タイプ  ISSN  
収録物識別子  00222488  
書誌レコードID  
収録物識別子タイプ  NCID  
収録物識別子  AA00701758  
DOI  
関連識別子  
識別子タイプ  DOI  
関連識別子  http://doi.org/10.1063/1.5021462  
関連名称  
関連名称  10.1063/1.5021462  
権利  
権利情報  © 2018 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. / The following article appeared in JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018) and may be found at (http://doi.org/10.1063/1.5021462).  
著者版フラグ  
値  author  
WoS  
表示名  Web of Science  
URL  http://gateway.isiknowledge.com/gateway/Gateway.cgi?&GWVersion=2&SrcAuth=ShinshuUniv&SrcApp=ShinshuUniv&DestLinkType=FullRecord&DestApp=WOS&KeyUT=000424017000042 