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

Special Matrices (2014)

- Volume: 2, Issue: 1, page 131-147, electronic only
- ISSN: 2300-7451

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topF. Štampach, and P. Šťovíček. "The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix." Special Matrices 2.1 (2014): 131-147, electronic only. <http://eudml.org/doc/266893>.

@article{F2014,

abstract = {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.},

author = {F. Štampach, P. Šťovíček},

journal = {Special Matrices},

keywords = {Jacobi matrix; Hahn-Exton q-Bessel function; self-adjoint extension; spectral problem; Hahn-Exton -Bessel function},

language = {eng},

number = {1},

pages = {131-147, electronic only},

title = {The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix},

url = {http://eudml.org/doc/266893},

volume = {2},

year = {2014},

}

TY - JOUR

AU - F. Štampach

AU - P. Šťovíček

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

JO - Special Matrices

PY - 2014

VL - 2

IS - 1

SP - 131

EP - 147, electronic only

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/266893

ER -

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