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{{DISPLAYTITLE:Jackson ''q''-Bessel function}}
In mathematics, a '''Jackson ''q''-Bessel function''' (or '''basic Bessel function''') is one of the three [[q-analog|''q''-analog]]s of the [[Bessel function]] introduced by {{harvs|txt|authorlink=F. H. Jackson|last=Jackson|year1=1903|year2=1903b|year3=1905|year4=1905b}}. The third Jackson ''q''-Bessel function is the same as the [[Hahn–Exton q-Bessel function|Hahn–Exton ''q''-Bessel function]].

==Definition==

The three Jackson ''q''-Bessel functions are given in terms of the [[q-Pochhammer symbol|''q''-Pochhammer symbol]] and the [[basic hypergeometric function]] <math>\phi</math> by

:<math> J_\nu^{(1)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_2\phi_1(0,0;q^{\nu+1};q,-x^2/4), \quad |x|<2, </math>
:<math> J_\nu^{(2)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_0\phi_1(;q^{\nu+1};q,-x^2q^{\nu +1}/4), \quad x\in\mathbb{C}, </math>
:<math> J_\nu^{(3)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_1\phi_1(0;q^{\nu+1};q,qx^2/4), \quad x\in\mathbb{C}.</math>
They can be reduced to the Bessel function by the continuous limit:<br>
:<math>\lim_{q\to1}J_\nu^{(k)}(x(1-q);q)=J_\nu(x), \ k=1,2,3.</math>
There is a connection formula between the first and second Jackson ''q''-Bessel function ({{harvtxt|Gasper|Rahman|2004}}):
:<math>J_\nu^{(2)}(x;q)=(-x^2/4;q)_\infty J_\nu^{(1)}(x;q), \ |x|<2.</math>
For integer order, the ''q''-Bessel functions satisfy
:<math>J_n^{(k)}(-x;q)=(-1)^n J_n^{(k)}(x;q), \ n\in\mathbb{Z}, \ k=1,2,3.</math>

==Properties==
===Negative Integer Order===
By using the relations ({{harvtxt|Gasper|Rahman|2004}}):
:<math>(q^{m+1};q)_\infty=(q^{m+n+1};q)_\infty (q^{m+1};q)_n,</math>
:<math>(q;q)_{m+n}=(q;q)_m (q^{m+1};q)_n,\ m,n\in\mathbb{Z},</math>
we obtain
:<math>J_{-n}^{(k)}(x;q)=(-1)^n J_n^{(k)}(x;q), \ k=1,2.</math>

===Zeros===
Hahn mentioned that <math> J_\nu^{(2)}(x;q)</math> has infinitely many real zeros ({{harvs|txt|authorlink=Wolfgang Hahn|last=Hahn|year=1949}}). Ismail proved that for <math>\nu>-1</math> all non-zero roots of <math> J_\nu^{(2)}(x;q)</math> are real ({{harvs|txt|authorlink=Mourad E. H. Ismail|last=Ismail|year=1982}}).

===Ratio of ''q''-Bessel Functions===
The function <math>-ix^{-1/2}J_{\nu+1}^{(2)}(ix^{1/2};q)/J_{\nu}^{(2)}(ix^{1/2};q)</math> is a [[completely monotonic function]] ({{harvs|txt|authorlink=Mourad E. H. Ismail|last=Ismail|year=1982}}).

===Recurrence Relations===
The first and second Jackson ''q''-Bessel function have the following recurrence relations (see {{harvs|txt|authorlink=Mourad E. H. Ismail|last=Ismail|year=1982}} and {{harvtxt|Gasper|Rahman|2004}}):
:<math>q^\nu J_{\nu+1}^{(k)}(x;q)=\frac{2(1-q^\nu)}{x}J_\nu^{(k)}(x;q)-J_{\nu-1}^{(k)}(x;q), \ k=1,2.</math>
:<math>J_{\nu}^{(1)}(x\sqrt{q};q)=q^{\pm\nu/2}\left(J_\nu^{(1)}(x;q)\pm \frac{x}{2}J_{\nu\pm1}^{(1)}(x;q)\right).</math>

===Inequalities===
When <math>\nu>-1</math>, the second Jackson ''q''-Bessel function satisfies:
<math>
\left|J_{\nu}^{(2)}(z;q)\right|\leq\frac{(-\sqrt{q};q)_{\infty}}{(q;q)_{\infty}}\left(\frac{|z|}{2}\right)^\nu\exp\left\{\frac{\log\left(|z|^2q^\nu/4\right)}{2\log q}\right\}.
</math>
(see {{harvs|txt|last=Zhang|year=2006}}.)

For <math> n\in\mathbb{Z}</math>, 
<math>
\left|J_{n}^{(2)}(z;q)\right|\leq\frac{(-q^{n+1};q)_{\infty}}{(q;q)_{\infty}}\left(\frac{|z|}{2}\right)^n(-|z|^2;q)_{\infty}.
</math>
(see {{harvs|txt|last=Koelink|year=1993}}.)

===Generating Function===
The following formulas are the ''q''-analog of the generating function for the Bessel function (see {{harvtxt|Gasper|Rahman|2004}}):<br>
:<math>\sum_{n=-\infty}^{\infty}t^nJ_n^{(2)}(x;q)=(-x^2/4;q)_{\infty}e_q(xt/2)e_q(-x/2t),</math>
:<math>\sum_{n=-\infty}^{\infty}t^nJ_n^{(3)}(x;q)=e_q(xt/2)E_q(-qx/2t).</math>
<math>e_q</math> is the [[q-exponential|''q''-exponential]] function.

==Alternative Representations==
===Integral Representations===
The second Jackson ''q''-Bessel function has the following integral representations (see {{harvs|txt|authorlink=Mizan Rahman|last=Rahman|year=1987}} and {{harvs|txt|authorlink1=Mourad E. H. Ismail|R. Zhang|last1=Ismail|last2=Zhang|year=2016}}):
:<math>
J_{\nu}^{(2)}(x;q)=\frac{(q^{2\nu};q)_{\infty}}{2\pi(q^{\nu};q)_{\infty}}(x/2)^{\nu} 
\cdot\int_0^{\pi} \frac{\left(e^{2i\theta}, e^{-2i\theta},-\frac{i x q^{(\nu+1)/2}}{2}e^{i\theta}, -\frac{i x q^{(\nu+1)/2}}{2}e^{-i\theta};q\right)_{\infty}}{(e^{2i\theta}q^{\nu}, e^{-2i\theta}q^{\nu};q)_{\infty}}\,d\theta,
</math>
:<math>
(a_1,a_2,\cdots,a_n;q)_{\infty}:=(a_1;q)_{\infty}(a_2;q)_{\infty}\cdots(a_n;q)_{\infty}, \ \Re \nu>0,
</math>
where <math>(a;q)_{\infty}</math>is the [[q-Pochhammer symbol|''q''-Pochhammer symbol]].  This representation reduces to the integral representation of the Bessel function in the limit <math>q\to 1</math>.
:<math>
J_{\nu}^{(2)}(z;q)=\frac{(z/2)^\nu}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\left(\frac{q^{\nu+1/2}z^2e^{ix}}{4};q\right)_{\infty}\exp\left(\frac{x^2}{\log q^2}\right)}{(q,-q^{\nu+1/2}e^{ix};q)_{\infty}}\,dx.
</math>

===Hypergeometric Representations===
The second Jackson ''q''-Bessel function has the following hypergeometric representations (see {{harvs|txt|last=Koelink|year=1993}}, {{harvs|txt|last1=Chen|authorlink2=Mourad Ismail|last2=Ismail|last3=Muttalib|year=1994}}):
:<math>
J_{\nu}^{(2)}(x;q)=\frac{(x/2)^{\nu}}{(q;q)_{\infty}}\ _1\phi_1(-x^2/4;0;q,q^{\nu+1}), 
</math>
:<math>
J_{\nu}^{(2)}(x;q)=\frac{(x/2)^{\nu}(\sqrt{q};q)_{\infty}}{2(q;q)_{\infty}}[f(x/2,q^{(\nu+1/2)/2};q)+f(-x/2,q^{(\nu+1/2)/2};q)], \ f(x,a;q):=(iax;\sqrt{q})_\infty \ _3\phi_2 \left(\begin{matrix} 
a, & -a, & 0 \\ 
-\sqrt{q}, & iax \end{matrix} 
; \sqrt{q},\sqrt{q} \right).
</math>
An asymptotic expansion can be obtained as an immediate consequence of the second formula.

For other hypergeometric representations, see {{harvs|txt|authorlink=Mizan Rahman|last=Rahman|year=1987}}.

==Modified ''q''-Bessel Functions==
The ''q''-analog of the modified Bessel functions are defined with the Jackson ''q''-Bessel function ({{harvs|txt|authorlink=Mourad Ismail|last=Ismail|year=1981}} and {{harvtxt|Olshanetsky|Rogov|1995}}):
:<math>I_\nu^{(j)}(x;q)=e^{i\nu\pi/2}J_{\nu}^{(j)}(x;q), \ j=1,2.</math>
:<math>K_\nu^{(j)}(x;q)=\frac{\pi}{2\sin(\pi\nu)}\left\{I_{-\nu}^{(j)}(x;q)-I_\nu^{(j)}(x;q)\right\}, \ j=1,2,\ \nu\in\mathbb{C}-\mathbb{Z},</math>
:<math>K_n^{(j)}(x;q)=\lim_{\nu\to n}K_\nu^{(j)}(x;q),\ n\in\mathbb{Z}.</math>
There is a connection formula between the modified q-Bessel functions:
:<math>I_\nu^{(2)}(x;q)=(-x^2/4;q)_\infty I_\nu^{(1)}(x;q).</math>
For statistical applications, see {{harvtxt|Kemp|1997}}.
===Recurrence Relations===
By the recurrence relation of Jackson ''q''-Bessel functions and the definition of modified ''q''-Bessel functions, the following recurrence relation can be obtained (<math>K_\nu^{(j)}(x;q)</math> also satisfies the same relation) ({{harvs|txt|authorlink=Mourad Ismail|last=Ismail|year=1981}}):
:<math>q^\nu I_{\nu+1}^{(j)}(x;q)=\frac{2}{z}(1-q^\nu)I_\nu^{(j)}(x;q)+I_{\nu-1}^{(j)}(x;q), \ j=1, 2.</math>
For other recurrence relations, see {{harvtxt|Olshanetsky|Rogov|1995}}.

===Continued Fraction Representation===
The ratio of modified ''q''-Bessel functions form a continued fraction ({{harvs|txt|authorlink=Mourad Ismail|last=Ismail|year=1981}}):
:<math>\frac{I_\nu^{(2)}(z;q)}{I_{\nu-1}^{(2)}(z;q)}=\cfrac{1}{2(1-q^\nu)/z+\cfrac{q^\nu}{2(1-q^{\nu+1})/z+\cfrac{q^{\nu+1}}{2(1-q^{\nu+2})/z+\ddots}}}.</math>

===Alternative Representations===
====Hypergeometric Representations====
The function <math>I_\nu^{(2)}(z;q)</math> has the following representation ({{harvtxt|Ismail|Zhang|2018}}):
:<math>
I_\nu^{(2)}(z;q)=\frac{(z/2)^\nu}{(q,q)_{\infty}} {}_1\phi_1(z^2/4;0;q,q^{\nu+1}).
</math>

====Integral Representations====
The modified ''q''-Bessel functions have the following integral representations ({{harvs|txt|authorlink=Mourad Ismail|last=Ismail|year=1981}}):
:<math>I_\nu^{(2)}(z;q)=\left(z^2/4;q\right)_\infty\left(\frac{1}{\pi}\int_0^\pi\frac{\cos\nu\theta\,d\theta}{\left(e^{i\theta}z/2;q\right)_\infty\left(e^{-i\theta}z/2;q\right)_\infty}-\frac{\sin\nu\pi}{\pi}\int_0^\infty\frac{e^{-\nu t}\,dt}{\left(-e^t z/2;q\right)_\infty\left(-e^{-t}z/2;q\right)_\infty}\right),</math>
:<math>K_\nu^{(1)}(z;q)=\frac{1}{2}\int_0^\infty\frac{e^{-\nu t}\,dt}{\left(-e^{t/2} z/2;q\right)_\infty\left(-e^{-t/2}z/2;q\right)_\infty},\ |\arg z|<\pi/2,</math>
:<math>K_\nu^{(1)}(z;q)=\int_0^\infty\frac{\cosh\nu \,dt}{\left(-e^{t/2} z/2;q\right)_\infty\left(-e^{-t/2}z/2;q\right)_\infty}.</math>

==See also==
*[[q-Bessel polynomials|''q''-Bessel polynomials]]

==References==

*{{Citation | last1=Ismail | first1=Mourad E. H. | title=The zeros of basic Bessel functions, the functions ''J''<sub>ν+''ax''</sub>(''x''), and associated orthogonal polynomials | doi=10.1016/0022-247X(82)90248-7 | mr=649849 | year=1982 | journal=[[Journal of Mathematical Analysis and Applications]] | issn=0022-247X | volume=86 | issue=1 | pages=1–19}}
*{{Citation | last1=Jackson | first1=F. H. | title=On generalized functions of Legendre and Bessel | year=1903 | journal= Transactions of the Royal Society of Edinburgh | volume=41 | pages=1–28| doi=10.1017/S0080456800080017 }}
*{{Citation | last1=Jackson | first1=F. H. | title= Theorems relating to a generalization of the Bessel functions | year=1903 | journal= Transactions of the Royal Society of Edinburgh | volume=41 | pages= 105–118| doi=10.1017/S0080456800080078 | url=https://zenodo.org/record/1804446 }}
*{{Citation | last1=Jackson | first1=F. H. | title=Theorems relating to a generalization of Bessel's function | jfm=36.0513.02 | year=1904 | journal=Transactions of the Royal Society of Edinburgh | volume=41 | issue=2 | pages=399–408 | doi=10.1017/s0080456800034475}}
*{{Citation | last1=Jackson | first1=F. H. | title=The Application of Basic Numbers to Bessel's and Legendre's Functions | doi=10.1112/plms/s2-2.1.192 | year=1905 | journal=Proceedings of the London Mathematical Society | volume=2 | issue=1 | pages=192–220| url=https://zenodo.org/record/1433525/files/article.pdf }}
*{{Citation | last1=Jackson | first1=F. H. | title=The Application of Basic Numbers to Bessel's and Legendre's Functions (Second paper) | doi=10.1112/plms/s2-3.1.1 | year=1905 | journal=Proceedings of the London Mathematical Society | volume=3 | issue=1 | pages=1–23| url=https://zenodo.org/record/1447790 }}
*{{Citation | last1=Rahman | first1=M | title=An Integral Representation and Some Transformation Properties of ''q''-Bessel Functions | year=1987 | journal= [[Journal of Mathematical Analysis and Applications]] | volume=125 | pages=58–71 | doi=10.1016/0022-247x(87)90164-8}}
*{{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 }}
*{{Citation | last1=Ismail | first1=M. E. H. | last2=Zhang|first2=R.|title=''q''-Bessel Functions and Rogers-Ramanujan Type Identities| year=2018 |journal=Proceedings of the American Mathematical Society | volume=146 | issue=9 | pages=3633–3646 |arxiv=1508.06861| doi=10.1090/proc/13078 }}
*{{Citation | last1=Koelink | first1=H. T. | title=Hansen-Lommel Orthogonality Relations for Jackson's ''q''-Bessel Functions | year=1993 | journal= [[Journal of Mathematical Analysis and Applications]] | volume=175 | issue=2 | pages=425–437 | doi=10.1006/jmaa.1993.1181}}
*{{cite arxiv |last1=Zhang|first1=R.|title=Plancherel-Rotach Asymptotics for ''q''-Series | year=2006 |eprint=math/0612216 |mode=cs2}}
*{{Citation | last1=Gasper | first1=G. | last2=Rahman | first2=M. | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
*{{Citation |last1=Chen|first1=Yang| last2=Ismail | first2=Mourad E. H. |last3=Muttalib|first3=K.A. | title=Asymptotics of basic Bessel functions and ''q''-Laguerre polynomials| year=1994 | journal=Journal of Computational and Applied Mathematics | volume=54 |issue=3| pages=263–272 | doi=10.1016/0377-0427(92)00128-v}}
*{{Citation | last1=Ismail | first1=Mourad | title=The Basic Bessel Functions and Polynomials | year=1981 | journal=SIAM Journal on Mathematical Analysis | volume=12 | issue=3 | pages=454–468 | doi=10.1137/0512038}}
*{{cite arxiv | last1=Olshanetsky |first1=M. A. |last2=Rogov |first2=V. B. |year=1995 |title=The Modified ''q''-Bessel Functions and the ''q''-Bessel-Macdonald Functions |eprint=q-alg/9509013 |mode=cs2}}

[[Category:Special functions]]
[[Category:Q-analogs]]