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 ({{harvs|txt|last=Swarttouw|year=1992}}). This function was introduced by {{harvs|txt|last=Hahn|authorlink=Wolfgang Hahn|year=1953}} in a special case and by {{harvs|txt|last=Exton|authorlink=Harold Exton|year=1983}} in general.

The Hahn–Exton q-Bessel function is given by

: $J_\nu^{(3)}(x;q) = \frac{x^\nu(q^{\nu+1};q)_\infty}{(q;q)_\infty} \sum_{k\ge 0}\frac{(-1)^kq^{k(k+1)/2}x^{2k}}{(q^{\nu+1};q)_k(q;q)_k}= \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} x^\nu {}_1\phi_1(0;q^{\nu+1};q,qx^2).$

$\phi$ is the basic hypergeometric function.

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 ({{harvs|txt|last1=Koelink|last2=Swarttouw|year=1994}}). For more details, see {{harvtxt|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 ({{harvtxt|Hahn|1953}}, {{harvtxt|Exton|1983}})

=Derivatives=

For the (usual) derivative and q-derivative of $J_\nu^{(3)}(x;q)$ , see {{harvs|txt|last1=Koelink|last2=Swarttouw|year=1994}}. The symmetric q-derivative of $J_\nu^{(3)}(x;q)$ is described on {{harvs|txt|last1=Cardoso|year=2016}}.

=Recurrence Relation=

The Hahn–Exton q-Bessel function has the following recurrence relation (see {{harvs|txt|last=Swarttouw|year=1992}}):

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

Alternative Representations

=Integral Representation=

The Hahn–Exton q-Bessel function has the following integral representation (see {{harvs|txt|authorlink1=Mourad E. H. Ismail|R. Zhang|last1=Ismail|last2=Zhang|year=2018}}):

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

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

=Hypergeometric Representation=

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see {{harvs|txt|last=Daalhuis|year=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

{{Citation | last1=Abreu | first1=L. D. | last2=Bustoz | first2=J. | last3=Cardoso | first3=J. L. |title=The Roots of the Third Jackson q-Bessel Function.| year=2003 |journal=International Journal of Mathematics and Mathematical Sciences | volume=2003 | issue=67 | pages=4241–4248 | doi=10.1155/S016117120320613X | doi-access=free | hdl=10316/110959 | hdl-access=free }}
{{Citation | last1=Cardoso | first1=J. L. |title=A Few Properties of the Third Jackson q-Bessel Function.| year=2016 |journal=Analysis Mathematica | volume=42 | issue=4 | pages=323–337 | doi=10.1007/s10476-016-0402-8 | s2cid=126278001 }}
{{Citation | last1=Daalhuis | first1=A. B. O. |title=Asymptotic Expansions for q-Gamma, q-Exponential, and q-Bessel functions.| year=1994 |journal=Journal of Mathematical Analysis and Applications | volume=186 | issue=3 | pages=896–913 | doi=10.1006/jmaa.1994.1339 | url=https://ir.cwi.nl/pub/2248 | doi-access=free }}
{{Citation | last1=Exton | first1=Harold | title=q-hypergeometric functions and applications | url=https://books.google.com/books?id=3kHvAAAAMAAJ | publisher=Ellis Horwood Ltd. | location=Chichester | series=Ellis Horwood Series: Mathematics and its Applications | isbn=978-0-85312-491-7 | mr=708496 | year=1983}}
{{Citation | last1=Hahn | first1=Wolfgang | authorlink=Wolfgang Hahn | title=Die mechanische Deutung einer geometrischen Differenzengleichung | language=German | doi=10.1002/zamm.19530330811 | zbl=0051.15502 | year=1953 | journal=Zeitschrift für Angewandte Mathematik und Mechanik | issn=0044-2267 | volume=33 | issue=8–9 | pages=270–272| bibcode=1953ZaMM...33..270H }}
{{Citation | last1=Ismail | first1=M. E. H. | last2=Zhang|first2=R.|title=Integral and Series Representations of q-Polynomials and Functions: Part I | year=2018 |journal=Analysis and Applications|volume=16| issue=2 |pages=209–281|arxiv=1604.08441| doi=10.1142/S0219530517500129 | s2cid=119142457 }}
{{Citation |last1=Koelink|first1=H. T. |last2=Swarttouw | first2=René F.| title=On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials| year=1994 | journal=Journal of Mathematical Analysis and Applications | volume=186 |issue=3 | pages=690–710|bibcode=1997math......3215K |arxiv=math/9703215 |doi=10.1006/jmaa.1994.1327 |s2cid=14382540 }}
{{Citation | last1=Swarttouw | first1=René F. | title=An addition theorem and some product formulas for the Hahn-Exton q-Bessel functions | doi=10.4153/CJM-1992-052-6 | mr=1178574 | year=1992 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=44 | issue=4 | pages=867–879| doi-access=free }}
{{Citation | last1=Swarttouw | first1=René F. | title=The Hahn-Exton q-Bessel function|url=https://repository.tudelft.nl/islandora/object/uuid%3A0bfd7d96-bb29-4846-8023-6242ce15e18f?collection=research | year=1992 | journal=PhD Thesis, Delft Technical University}}

Category:Special functions

Category:Q-analogs