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 (Swarttouw (1992)). This function was introduced by Hahn (1953) in a special case and by Exton (1983) in general.

The Hahn–Exton q-Bessel function is given by

$J_{\nu }^{(3)}(x;q)=\frac{x^{\nu }(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}\sum _{k\ge 0}\frac{(-1{)}^k{q}^{k(k+1)/2}{x}^{2k}}{(q^{\nu +1};q{)}_k(q;q{)}_k}=\frac{(q^{\nu +1};q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{x}^{\nu }{}_1{\varphi }_1(0;q^{\nu +1};q,q{x}^2).$

$\varphi$ is the basic hypergeometric function.

Properties

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 (Koelink and Swarttouw (1994)). For more details, see Abreu, Bustoz & Cardoso (2003). 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 (Hahn (1953), Exton (1983))

Derivatives

For the (usual) derivative and q-derivative of $J_{\nu }^{(3)}(x;q)$, see Koelink and Swarttouw (1994). The symmetric q-derivative of $J_{\nu }^{(3)}(x;q)$ is described on Cardoso (2016).

Recurrence Relation

The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)):

$J_{\nu +1}^{(3)}(x;q)=(\frac{1-q^{\nu }}{x}+x)J_{\nu }^{(3)}(x;q)-J_{\nu -1}^{(3)}(x;q).$

Alternative Representations

Integral Representation

The Hahn–Exton q-Bessel function has the following integral representation (see Ismail and Zhang (2018)):

$J_{\nu }^{(3)}(z;q)=\frac{z^{\nu }}{\sqrt{\pi \log q^{-2}}}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\exp \left(\frac{x^2}{\log q^2}\right)}{(q,-q^{\nu +1/2}{e}^{ix},-q^{1/2}{z}^2{e}^{ix};q{)}_{\mathrm{\infty }}}dx.$

$(a_1,a_2,\cdots ,a_n;q{)}_{\mathrm{\infty }}:=(a_1;q{)}_{\mathrm{\infty }}(a_2;q{)}_{\mathrm{\infty }}\cdots (a_n;q{)}_{\mathrm{\infty }}.$

Hypergeometric Representation

The Hahn–Exton q-Bessel function has the following hypergeometric representation (see Daalhuis (1994)):

$J_{\nu }^{(3)}(x;q)=x^{\nu }\frac{(x^{2}q;q{)}_{\mathrm{\infty }}}{(q;q{)}_{\mathrm{\infty }}}{}_1{\varphi }_1(0;x^{2}q;q,q^{\nu +1}).$

This converges fast at $x\to \mathrm{\infty }$. It is also an asymptotic expansion for $\nu \to \mathrm{\infty }$.