Jackson q-Bessel function

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In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by Jackson (1906a, 1906b, 1905a, 1905b). The third Jackson q-Bessel function is the same as the Hahn–Exton q-Bessel function.

Definition

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

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,

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} ,

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:

\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)):

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

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

Properties

Negative Integer Order

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

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

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

we obtain

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

Zeros

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

Ratio of q-Bessel Functions

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

Recurrence Relations

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

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.

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).

Inequalities

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

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

Generating Function

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

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

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

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

Alternative Representations

Integral Representations

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

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 ,

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

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.

Hypergeometric Representations

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

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

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).

Modified q-Bessel Functions

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

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

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} ,

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:

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

For statistical applications, see Kemp (1997).