Jackson q-Bessel function

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).

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)):

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

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

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