Jackson q-Bessel function

The three Jackson q-Bessel functions are given in terms of the q-Pochhammer symbol and the basic hypergeometric function ${\displaystyle \phi }$  by

${\displaystyle J_{\nu }^{(1)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{2}\phi _{1}(0,0;q^{\nu +1};q,-x^{2}/4),\quad |x|<2,}$ 

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{0}\phi _{1}(;q^{\nu +1};q,-x^{2}q^{\nu +1}/4),\quad x\in \mathbb {C} ,}$ 

${\displaystyle J_{\nu }^{(3)}(x;q)={\frac {(q^{\nu +1};q)_{\infty }}{(q;q)_{\infty }}}(x/2)^{\nu }{}_{1}\phi _{1}(0;q^{\nu +1};q,qx^{2}/4),\quad x\in \mathbb {C} .}$ 

They can be reduced to the Bessel function by the continuous limit:

${\displaystyle \lim _{q\to 1}J_{\nu }^{(k)}(x(1-q);q)=J_{\nu }(x),\ k=1,2,3.}$ 

There is a connection formula between the first and second Jackson q-Bessel function (Gasper & Rahman (2004)):

${\displaystyle J_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }J_{\nu }^{(1)}(x;q),\ |x|<2.}$ 

For integer order, the q-Bessel functions satisfy

${\displaystyle J_{n}^{(k)}(-x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ n\in \mathbb {Z} ,\ k=1,2,3.}$ 

By using the relations (Gasper & Rahman (2004)):

${\displaystyle (q^{m+1};q)_{\infty }=(q^{m+n+1};q)_{\infty }(q^{m+1};q)_{n},}$ 

${\displaystyle (q;q)_{m+n}=(q;q)_{m}(q^{m+1};q)_{n},\ m,n\in \mathbb {Z} ,}$ 

we obtain

${\displaystyle J_{-n}^{(k)}(x;q)=(-1)^{n}J_{n}^{(k)}(x;q),\ k=1,2.}$ 

Hahn mentioned that ${\displaystyle J_{\nu }^{(2)}(x;q)}$  has infinitely many real zeros (Hahn (1949)). Ismail proved that for ${\displaystyle \nu >-1}$  all non-zero roots of ${\displaystyle J_{\nu }^{(2)}(x;q)}$  are real (Ismail (1982)).

The function ${\displaystyle -ix^{-1/2}J_{\nu +1}^{(2)}(ix^{1/2};q)/J_{\nu }^{(2)}(ix^{1/2};q)}$  is a completely monotonic function (Ismail (1982)).

The first and second Jackson q-Bessel function have the following recurrence relations (see Ismail (1982) and Gasper & Rahman (2004)):

${\displaystyle q^{\nu }J_{\nu +1}^{(k)}(x;q)={\frac {2(1-q^{\nu })}{x}}J_{\nu }^{(k)}(x;q)-J_{\nu -1}^{(k)}(x;q),\ k=1,2.}$ 

${\displaystyle J_{\nu }^{(1)}(x{\sqrt {q}};q)=q^{\pm \nu /2}\left(J_{\nu }^{(1)}(x;q)\pm {\frac {x}{2}}J_{\nu \pm 1}^{(1)}(x;q)\right).}$ 

When ${\displaystyle \nu >-1}$ , the second Jackson q-Bessel function satisfies: ${\displaystyle \left|J_{\nu }^{(2)}(z;q)\right|\leq {\frac {(-{\sqrt {q}};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{\nu }\exp \left\{{\frac {\log \left(|z|^{2}q^{\nu }/4\right)}{2\log q}}\right\}.}$  (see Zhang (2006).)

For ${\displaystyle n\in \mathbb {Z} }$ , ${\displaystyle \left|J_{n}^{(2)}(z;q)\right|\leq {\frac {(-q^{n+1};q)_{\infty }}{(q;q)_{\infty }}}\left({\frac {|z|}{2}}\right)^{n}(-|z|^{2};q)_{\infty }.}$  (see Koelink (1993).)

The following formulas are the q-analog of the generating function for the Bessel function (see Gasper & Rahman (2004)):

${\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }e_{q}(xt/2)e_{q}(-x/2t),}$ 

${\displaystyle \sum _{n=-\infty }^{\infty }t^{n}J_{n}^{(3)}(x;q)=e_{q}(xt/2)E_{q}(-qx/2t).}$ 

${\displaystyle e_{q}}$  is the q-exponential function.

The second Jackson q-Bessel function has the following integral representations (see Rahman (1987) and Ismail & Zhang (2018a)):

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(q^{2\nu };q)_{\infty }}{2\pi (q^{\nu };q)_{\infty }}}(x/2)^{\nu }\cdot \int _{0}^{\pi }{\frac {\left(e^{2i\theta },e^{-2i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{i\theta },-{\frac {ixq^{(\nu +1)/2}}{2}}e^{-i\theta };q\right)_{\infty }}{(e^{2i\theta }q^{\nu },e^{-2i\theta }q^{\nu };q)_{\infty }}}\,d\theta ,}$ 

${\displaystyle (a_{1},a_{2},\cdots ,a_{n};q)_{\infty }:=(a_{1};q)_{\infty }(a_{2};q)_{\infty }\cdots (a_{n};q)_{\infty },\ \Re \nu >0,}$ 

where ${\displaystyle (a;q)_{\infty }}$  is the q-Pochhammer symbol. This representation reduces to the integral representation of the Bessel function in the limit ${\displaystyle q\to 1}$ .

${\displaystyle J_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{\sqrt {2\pi \log q^{-1}}}}\int _{-\infty }^{\infty }{\frac {\left({\frac {q^{\nu +1/2}z^{2}e^{ix}}{4}};q\right)_{\infty }\exp \left({\frac {x^{2}}{\log q^{2}}}\right)}{(q,-q^{\nu +1/2}e^{ix};q)_{\infty }}}\,dx.}$ 

The second Jackson q-Bessel function has the following hypergeometric representations (see Koelink (1993), Chen, Ismail, and Muttalib (1994)):

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }}{(q;q)_{\infty }}}\ _{1}\phi _{1}(-x^{2}/4;0;q,q^{\nu +1}),}$ 

${\displaystyle J_{\nu }^{(2)}(x;q)={\frac {(x/2)^{\nu }({\sqrt {q}};q)_{\infty }}{2(q;q)_{\infty }}}[f(x/2,q^{(\nu +1/2)/2};q)+f(-x/2,q^{(\nu +1/2)/2};q)],\ f(x,a;q):=(iax;{\sqrt {q}})_{\infty }\ _{3}\phi _{2}\left({\begin{matrix}a,&-a,&0\\-{\sqrt {q}},&iax\end{matrix}};{\sqrt {q}},{\sqrt {q}}\right).}$ 

An asymptotic expansion can be obtained as an immediate consequence of the second formula.

For other hypergeometric representations, see Rahman (1987).

The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function (Ismail (1981) and Olshanetsky & Rogov (1995)):

${\displaystyle I_{\nu }^{(j)}(x;q)=e^{i\nu \pi /2}J_{\nu }^{(j)}(x;q),\ j=1,2.}$ 

${\displaystyle K_{\nu }^{(j)}(x;q)={\frac {\pi }{2\sin(\pi \nu )}}\left\{I_{-\nu }^{(j)}(x;q)-I_{\nu }^{(j)}(x;q)\right\},\ j=1,2,\ \nu \in \mathbb {C} -\mathbb {Z} ,}$ 

${\displaystyle K_{n}^{(j)}(x;q)=\lim _{\nu \to n}K_{\nu }^{(j)}(x;q),\ n\in \mathbb {Z} .}$ 

There is a connection formula between the modified q-Bessel functions:

${\displaystyle I_{\nu }^{(2)}(x;q)=(-x^{2}/4;q)_{\infty }I_{\nu }^{(1)}(x;q).}$ 

For statistical applications, see Kemp (1997).

By the recurrence relation of Jackson q-Bessel functions and the definition of modified q-Bessel functions, the following recurrence relation can be obtained (${\displaystyle K_{\nu }^{(j)}(x;q)}$  also satisfies the same relation) (Ismail (1981)):

${\displaystyle q^{\nu }I_{\nu +1}^{(j)}(x;q)={\frac {2}{z}}(1-q^{\nu })I_{\nu }^{(j)}(x;q)+I_{\nu -1}^{(j)}(x;q),\ j=1,2.}$ 

For other recurrence relations, see Olshanetsky & Rogov (1995).

The ratio of modified q-Bessel functions form a continued fraction (Ismail (1981)):

${\displaystyle {\frac {I_{\nu }^{(2)}(z;q)}{I_{\nu -1}^{(2)}(z;q)}}={\cfrac {1}{2(1-q^{\nu })/z+{\cfrac {q^{\nu }}{2(1-q^{\nu +1})/z+{\cfrac {q^{\nu +1}}{2(1-q^{\nu +2})/z+\ddots }}}}}}.}$ 

The function ${\displaystyle I_{\nu }^{(2)}(z;q)}$  has the following representation (Ismail & Zhang (2018b)):

${\displaystyle I_{\nu }^{(2)}(z;q)={\frac {(z/2)^{\nu }}{(q,q)_{\infty }}}{}_{1}\phi _{1}(z^{2}/4;0;q,q^{\nu +1}).}$ 

The modified q-Bessel functions have the following integral representations (Ismail (1981)):

${\displaystyle I_{\nu }^{(2)}(z;q)=\left(z^{2}/4;q\right)_{\infty }\left({\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\cos \nu \theta \,d\theta }{\left(e^{i\theta }z/2;q\right)_{\infty }\left(e^{-i\theta }z/2;q\right)_{\infty }}}-{\frac {\sin \nu \pi }{\pi }}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t}z/2;q\right)_{\infty }\left(-e^{-t}z/2;q\right)_{\infty }}}\right),}$ 

${\displaystyle K_{\nu }^{(1)}(z;q)={\frac {1}{2}}\int _{0}^{\infty }{\frac {e^{-\nu t}\,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}},\ |\arg z|<\pi /2,}$ 

${\displaystyle K_{\nu }^{(1)}(z;q)=\int _{0}^{\infty }{\frac {\cosh \nu \,dt}{\left(-e^{t/2}z/2;q\right)_{\infty }\left(-e^{-t/2}z/2;q\right)_{\infty }}}.}$ 

• q-Bessel polynomials

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