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

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

$\phi$ is the basic hypergeometric function.

Zeros

Koelink and Swarttouw proved that $J_{\nu }^{(3)}(x;q)$ has infinite number of real zeros (Koelink and Swarttouw (1994)).

Recurrence Relation

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

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

Integral Representation

The Hahn-Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2016)):

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

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

Hypergeometric Representation

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

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

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

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: 270–272, 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
• Ismail, Mourad E. H.; Zhang, R. (2016), "Integral and Series Representations of q-Polynomials and Functions: Part I", arXiv:1604.08441 [math.CA]
• 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