# Q-Bessel functions

The three Jackson $q$-Bessel functions are $$J_{\nu}^{(1)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_2\Phi_1\left(0,0;q^{\nu+1};q,-\dfrac{x^2}{4} \right),$$ $$J_{\nu}^{(2)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_0\Phi_1\left(-q^{\nu+1};q,-\dfrac{x^2q^{\nu+1}}{4} \right),$$ and $$J_{\nu}^{(3)}(x;q) = \dfrac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}} \left( \dfrac{x}{2} \right)^{\nu} {}_1\Phi_1\left(0;q^{\nu+1};q,\dfrac{qx^2}{4} \right),$$ where $(q^{\nu+1};q)_{\infty}$ and $(q,q)_{\infty}$ are q-Pochhammer symbols and ${}_p\Phi_q$ denotes the hypergeometric $q$-series