Jackson q-Bessel function

In mathematics, a Jackson q-Bessel function (or basic Bessel function) is one of the three q-analogs of the Bessel function introduced by. 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


 * $$ J_\nu^{(1)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_2\phi_1(0,0;q^{\nu+1};q,-x^2/4), \quad |x|<2, $$
 * $$ J_\nu^{(2)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_0\phi_1(q^{\nu+1};q,-x^2q^{\nu +1}/4), \quad x\in\mathbb{C}, $$
 * $$ J_\nu^{(3)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_1\phi_1(0;q^{\nu+1};q,qx^2/4), \quad x\in\mathbb{C}.$$

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

For integer order, the q-Bessel functions satisfy
 * $$J_n^{(k)}(-x;q)=(-1)^n J_n^{(k)}(x;q), \ n\in\mathbb{Z}, \ k=1,2,3.$$

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.

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 \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^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} $$ An asymptotic expansion can be obtained as an immediate consequence of the second formula.
 * \sqrt{q},\sqrt{q} \right).

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

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 :
 * $$I_\nu^{(2)}(z;q)=\left(z^2/4;q\right)_\infty\left(\frac{1}{\pi}\int_0^\pi\frac{\cos\nu\theta\,d\theta}{\left(e^{i\theta}z/2;q\right)_\infty\left(e^{-i\theta}z/2;q\right)_\infty}-\frac{\sin\nu\pi}{\pi}\int_0^\infty\frac{e^{-\nu t}\,dt}{\left(-e^t z/2;q\right)_\infty\left(-e^{-t}z/2;q\right)_\infty}\right),$$
 * $$K_\nu^{(1)}(z;q)=\frac{1}{2}\int_0^\infty\frac{e^{-\nu t}\,dt}{\left(-e^{t/2} z/2;q\right)_\infty\left(-e^{-t/2}z/2;q\right)_\infty},\ |\arg z|<\pi/2,$$
 * $$K_\nu^{(1)}(z;q)=\int_0^\infty\frac{\cosh\nu \,dt}{\left(-e^{t/2} z/2;q\right)_\infty\left(-e^{-t/2}z/2;q\right)_\infty}.$$