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Special Matrices

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The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix

F. Štampach
  • Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Kolejní 2, 16000 Praha, Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ P. Šťovíček
  • Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in Prague, Trojanova 13, 12000 Praha, Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-10-01 | DOI: https://doi.org/10.2478/spma-2014-0014


A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton q-Bessel function Jν(z; q) serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the q-Bessel function due to Koelink and Swarttouw.

Keywords: Jacobi matrix; Hahn-Exton q-Bessel function; self-adjoint extension; spectral problem


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About the article

Received: 2014-02-05

Accepted: 2014-09-14

Published Online: 2014-10-01

Citation Information: Special Matrices, Volume 2, Issue 1, ISSN (Online) 2300-7451, DOI: https://doi.org/10.2478/spma-2014-0014.

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© 2014 F. Štampach, P. Šťovíček. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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