Hahn–Exton q-Bessel function
In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation (Swarttouw (1992)). This function was introduced by Hahn (1953) in a special case and by Exton (1983) in general.
The Hahn–Exton q-Bessel function is given by
is the basic hypergeometric function.
Properties
Zeros
Koelink and Swarttouw proved that has infinite number of real zeros (Koelink and Swarttouw (1994)).
Derivative
For the (usual) derivative of , see Koelink and Swarttouw (1994).
Recurrence Relation
The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)):
Alternative Representations
Integral Representation
The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2016)):
Hypergeometric Representation
The Hahn–Exton q-Bessel function has the following hypergeometric representation (see Daalhuis (1994)):
This converges fast at . It is also an asymptotic expansion for .
References
- Exton, Harold (1983), q-hypergeometric functions and applications, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-491-7, MR 0708496
- Hahn, Wolfgang (1953), "Die mechanische Deutung einer geometrischen Differenzengleichung", Zeitschrift für Angewandte Mathematik und Mechanik (in German), 33 (8–9): 270–272, Bibcode:1953ZaMM...33..270H, doi:10.1002/zamm.19530330811, ISSN 0044-2267, Zbl 0051.15502
- Ismail, Mourad. E. H. (2003), Some Properties of Jacksons Third q-Bessel Function, preprint
- Swarttouw, René F. (1992), "An addition theorem and some product formulas for the Hahn-Exton q-Bessel functions", Canadian Journal of Mathematics, 44 (4): 867–879, doi:10.4153/CJM-1992-052-6, ISSN 0008-414X, MR 1178574
- Swarttouw, René F. (1992), "The Hahn-Exton q-Bessel function", PhD thesis, Delft Technical University
- Koelink, H. T.; Swarttouw, René F. (1994), "On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials", Journal of Mathematical Analysis and Applications, 186: 690–710, arXiv:math/9703215, Bibcode:1997math......3215K
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ignored (help) - Ismail, Mourad E. H.; Zhang, R. (2016), Integral and Series Representations of q-Polynomials and Functions: Part I, arXiv:1604.08441
- Daalhuis, A. B. O. (1994), "Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions.", Journal of Mathematical Analysis and Applications, 186 (3): 896–913