Jackson 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| : 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}, : 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\to1}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 (): :J_\nu^{(2)}(x;q)=(-x^2/4;q)\infty J\nu^{(1)}(x;q), \ |x| 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 (): :(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 (). Ismail proved that for \nu-1 all non-zero roots of J_\nu^{(2)}(x;q) are real ().

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 ().

Recurrence Relations

The first and second Jackson q-Bessel function have the following recurrence relations (see and ): :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\pm1}^{(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|^2q^\nu/4\right)}{2\log q}\right}. (see .)

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 .)

Generating Function

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

:\sum_{n=-\infty}^{\infty}t^nJ_n^{(2)}(x;q)=(-x^2/4;q){\infty}e_q(xt/2)e_q(-x/2t), :\sum{n=-\infty}^{\infty}t^nJ_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 and ): : 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, : (a_1,a_2,\cdots,a_n;q){\infty}:=(a_1;q){\infty}(a_2;q){\infty}\cdots(a_n;q){\infty}, \ \Re \nu0, 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^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.

Hypergeometric Representations

The second Jackson q-Bessel function has the following hypergeometric representations (see , ): : 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 .

Modified ''q''-Bessel Functions

The q-analog of the modified Bessel functions are defined with the Jackson q-Bessel function ( and ): :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 .

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 (K_\nu^{(j)}(x;q) also satisfies the same relation) (): :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 .

Continued Fraction Representation

The ratio of modified q-Bessel functions form a continued fraction (): :\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}}}.

Alternative Representations

Hypergeometric Representations

The function I_\nu^{(2)}(z;q) has the following representation (): : I_\nu^{(2)}(z;q)=\frac{(z/2)^\nu}{(q,q)_{\infty}} {}_1\phi_1(z^2/4;0;q,q^{\nu+1}).

Integral Representations

The modified q-Bessel functions have the following integral representations (): :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), :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| :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}.

References