Hahn–Exton q-Bessel function

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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

[math]\displaystyle{ 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). }[/math]

[math]\displaystyle{ \phi }[/math] is the basic hypergeometric function.

Properties

Zeros

Koelink and Swarttouw proved that [math]\displaystyle{ J_\nu^{(3)}(x;q) }[/math] has infinite number of real zeros. They also proved that for [math]\displaystyle{ \nu\gt -1 }[/math] all non-zero roots of [math]\displaystyle{ J_\nu^{(3)}(x;q) }[/math] are real (Koelink and Swarttouw (1994)). For more details, see (Abreu Bustoz). 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 [math]\displaystyle{ J_\nu^{(3)}(x;q) }[/math], see Koelink and Swarttouw (1994). The symmetric q-derivative of [math]\displaystyle{ J_\nu^{(3)}(x;q) }[/math] is described on Cardoso (2016).

Recurrence Relation

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

[math]\displaystyle{ 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). }[/math]

Alternative Representations

Integral Representation

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

[math]\displaystyle{ 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. }[/math]
[math]\displaystyle{ (a_1,a_2,\cdots,a_n;q)_{\infty}:=(a_1;q)_{\infty}(a_2;q)_{\infty}\cdots(a_n;q)_{\infty}. }[/math]

Hypergeometric Representation

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

[math]\displaystyle{ 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}). }[/math]

This converges fast at [math]\displaystyle{ x\to\infty }[/math]. It is also an asymptotic expansion for [math]\displaystyle{ \nu\to\infty }[/math].

References



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