Hahn–Exton q-Bessel function

(Redirected from Hahn-Exton 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

edit

Zeros

edit

Koelink and Swarttouw proved that   has infinite number of real zeros. They also proved that for   all non-zero roots of   are real (Koelink and Swarttouw (1994)). For more details, see Abreu, Bustoz & Cardoso (2003). Zeros of the Hahn-Exton q-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (Hahn (1953), Exton (1983))

Derivatives

edit

For the (usual) derivative and q-derivative of  , see Koelink and Swarttouw (1994). The symmetric q-derivative of   is described on Cardoso (2016).

Recurrence Relation

edit

The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)):

 

Alternative Representations

edit

Integral Representation

edit

The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2018)):

 
 

Hypergeometric Representation

edit

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

edit
  • Abreu, L. D.; Bustoz, J.; Cardoso, J. L. (2003), "The Roots of the Third Jackson q-Bessel Function.", International Journal of Mathematics and Mathematical Sciences, 2003 (67): 4241–4248, doi:10.1155/S016117120320613X, hdl:10316/110959
  • Cardoso, J. L. (2016), "A Few Properties of the Third Jackson q-Bessel Function.", Analysis Mathematica, 42 (4): 323–337, doi:10.1007/s10476-016-0402-8, S2CID 126278001
  • 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, doi:10.1006/jmaa.1994.1339
  • 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, M. E. H.; Zhang, R. (2018), "Integral and Series Representations of q-Polynomials and Functions: Part I", Analysis and Applications, 16 (2): 209–281, arXiv:1604.08441, doi:10.1142/S0219530517500129, S2CID 119142457
  • 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 (3): 690–710, arXiv:math/9703215, Bibcode:1997math......3215K, doi:10.1006/jmaa.1994.1327, S2CID 14382540
  • 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