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 Jackson (1903, 1903b, 1905, 1905b). 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 Pochhammer symbol and the basic hypergeometric function φ by

$J_\nu^{(1)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_2\phi_1(0,0;q^{\nu+1};q,-x^2/4)$

$J_\nu^{(2)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_0\phi_1(;q^{\nu+1};q,-x^2q^{\nu +1}/4)$

$J_\nu^{(3)}(x;q) = \frac{(q^{\nu+1};q)_\infty}{(q;q)_\infty} (x/2)^\nu {}_1\phi_1(0;q^{\nu+1};q,qx^2/4)$