Hahn–Exton q-Bessel 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 (). 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).

Alternative Representations

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.

References