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

$J_{\nu }^{(3)}(x;q)=\frac{x^{\nu }(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}\sum _{k\ge 0}\frac{(-1{)}^k{q}^{k(k+1)/2}{x}^{2k}}{(q^{\nu +1};q{)}_k(q;q{)}_k}=\frac{(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{x}^{\nu }{}_1{\varphi }_1(0;q^{\nu +1};q,q{x}^2).$

$\varphi$ is the basic hypergeometric function.

Properties

Zeros

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

Recurrence Relation

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

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

Alternative Representations

Integral Representation

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

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

$(a_1,a_2,\cdots ,a_n;q{)}_{\mathrm{\infty }}:=(a_1;q{)}_{\mathrm{\infty }}(a_2;q{)}_{\mathrm{\infty }}\cdots (a_n;q{)}_{\mathrm{\infty }}.$

Hypergeometric Representation

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

$J_{\nu }^{(3)}(x;q)=x^{\nu }\frac{(x^{2}q;q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{}_1{\varphi }_1(0;x^{2}q;q,q^{\nu +1}).$

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

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 
• 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 
• 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, https://ir.cwi.nl/pub/2248 
• 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, https://books.google.com/books?id=3kHvAAAAMAAJ 
• Hahn, Wolfgang (1953), "Die mechanische Deutung einer geometrischen Differenzengleichung" (in German), Zeitschrift für Angewandte Mathematik und Mechanik 33 (8–9): 270–272, doi:10.1002/zamm.19530330811, ISSN 0044-2267, Bibcode: 1953ZaMM...33..270H 
• 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, doi:10.1142/S0219530517500129 
• 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, doi:10.1006/jmaa.1994.1327, Bibcode: 1997math......3215K 
• 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 
• Swarttouw, René F. (1992), "The Hahn-Exton q-Bessel function", PHD Thesis, Delft Technical University, https://repository.tudelft.nl/islandora/object/uuid%3A0bfd7d96-bb29-4846-8023-6242ce15e18f?collection=research