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

${\displaystyle J_{\nu }^{(3)}(x;q)={\frac {x^{\nu }(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}\sum _{k\geq 0}{\frac {(-1)^{k}q^{k(k+1)/2}x^{2k}}{(q^{\nu +1};q)_{k}(q;q)_{k}}}={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}x^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}).}$

${\displaystyle \phi }$ is the basic hypergeometric function.

Properties

Zeros

Koelink and Swarttouw proved that ${\displaystyle J_{\nu }^{(3)}(x;q)}$ has infinite number of real zeros. They also proved that for ${\displaystyle \nu >-1}$ all non-zero roots of ${\displaystyle J_{\nu }^{(3)}(x;q)}$ 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 ${\displaystyle J_{\nu }^{(3)}(x;q)}$, see Koelink and Swarttouw (1994). The symmetric q-derivative of ${\displaystyle J_{\nu }^{(3)}(x;q)}$ is described on Cardoso (2016).

Recurrence Relation

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

${\displaystyle J_{\nu +1}^{(3)}(x;q)=\left({\frac {1-q^{\nu }}{x}}+x\right)J_{\nu }^{(3)}(x;q)-J_{\nu -1}^{(3)}(x;q).}$

Alternative Representations

Integral Representation

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

${\displaystyle J_{\nu }^{(3)}(z;q)={\frac {z^{\nu }}{\sqrt {\pi \log q^{-2}}}}\int _{-\infty }^{\infty }{\frac {\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix},-q^{1/2}z^{2}e^{ix};q)_{\infty }}}\,dx.}$

${\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty }.}$

Hypergeometric Representation

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see Daalhuis (1994)):

${\displaystyle J_{\nu }^{(3)}(x;q)=x^{\nu }{\frac {(x^{2}q;q)_{\infty }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(0;x^{2}q;q,q^{\nu +1}).}$

This converges fast at ${\displaystyle x\to \infty }$. It is also an asymptotic expansion for ${\displaystyle \nu \to \infty }$.

References

• 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