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
Properties
Zeros
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
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
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
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
- Abreu, L. D.; Bustoz, J.; Cardoso, J. L. (2003), "The Roots of the Third Jackson q-Bessel Function.", hdl:10316/110959
- Cardoso, J. L. (2016), "A Few Properties of the Third Jackson q-Bessel Function.", S2CID 126278001
- Daalhuis, A. B. O. (1994), "Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions.",
- Exton, Harold (1983), q-hypergeometric functions and applications, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., MR 0708496
- 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, S2CID 119142457
- Koelink, H. T.; Swarttouw, René F. (1994), "On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials", S2CID 14382540
- Swarttouw, René F. (1992), "An addition theorem and some product formulas for the Hahn-Exton q-Bessel functions", MR 1178574
- Swarttouw, René F. (1992), "The Hahn-Exton q-Bessel function", PHD Thesis, Delft Technical University