2024-03-29T08:28:25Z
https://soar-ir.repo.nii.ac.jp/oai
oai:soar-ir.repo.nii.ac.jp:00019826
2022-12-14T04:16:24Z
1169:1170
Orthogonal polynomials from Hermitian matrices. II
Odake, Satoru
Sasaki, Ryu
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 q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness 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 two-unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the '2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two-component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) polynomials. Published by AIP Publishing.
Article
JOURNAL OF MATHEMATICAL PHYSICS. 59(1):013504 (2018)
journal article
AMER INST PHYSICS
2018-01
application/pdf
JOURNAL OF MATHEMATICAL PHYSICS
1
59
013504
0022-2488
AA00701758
https://soar-ir.repo.nii.ac.jp/record/19826/files/1604.00714v2.pdf
eng
10.1063/1.5021462
https://doi.org/10.1063/1.5021462
© 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 (https://doi.org/10.1063/1.5021462).