Hahn–Exton q-Bessel function

In mathematics, the Hahn–Exton q-Bessel function or the third Jackson q-Bessel function is a q-analog of the Bessel function, and satisfies the Hahn-Exton q-difference equation. This function was introduced by in a special case and by  in general.

The Hahn–Exton q-Bessel function is given by
 * $$ J_\nu^{(3)}(x;q) = \frac{x^\nu(q^{\nu+1};q)_\infty}{(q;q)_\infty} \sum_{k\ge 0}\frac{(-1)^kq^{k(k+1)/2}x^{2k}}{(q^{\nu+1};q)_k(q;q)_k}= \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} x^\nu {}_1\phi_1(0;q^{\nu+1};q,qx^2).$$

$$\phi$$ is the basic hypergeometric function.

Zeros
Koelink and Swarttouw proved that $$ J_\nu^{(3)}(x;q)$$ has infinite number of real zeros. They also proved that for $$\nu>-1$$ all non-zero roots of $$ J_\nu^{(3)}(x;q)$$ are real. For more details, see. Zeros of the Hahn-Exton q-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain

Derivatives
For the (usual) derivative and q-derivative of $$ J_\nu^{(3)}(x;q)$$, see. The symmetric q-derivative of $$ J_\nu^{(3)}(x;q)$$ is described on.

Recurrence Relation
The Hahn–Exton q-Bessel function has the following recurrence relation (see ):
 * $$ J_{\nu+1}^{(3)}(x;q)=\left(\frac{1-q^\nu}{x}+x\right)J_\nu^{(3)}(x;q)-J_{\nu-1}^{(3)}(x;q).$$

Integral Representation
The Hahn–Exton q-Bessel function has the following integral representation (see ):

J_{\nu}^{(3)}(z;q)=\frac{z^\nu}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^2}{\log q^2}\right)}{(q,-q^{\nu+1/2}e^{ix},-q^{1/2}z^2e^{ix};q)_{\infty}}\,dx. $$
 * $$(a_1,a_2,\cdots,a_n;q)_{\infty}:=(a_1;q)_{\infty}(a_2;q)_{\infty}\cdots(a_n;q)_{\infty}.$$

Hypergeometric Representation
The Hahn–Exton q-Bessel function has the following hypergeometric representation (see ):

J_{\nu}^{(3)}(x;q)=x^{\nu}\frac{(x^2 q;q)_{\infty}}{(q;q)_{\infty}}\ _1\phi_1(0;x^2 q;q,q^{\nu+1}). $$ This converges fast at $$x\to\infty$$. It is also an asymptotic expansion for $$\nu\to\infty$$.