Generalized q-Bessel

The generalized $q$-Bessel is defined by $$J_n(x,a;q) = \dfrac{(\frac{x}{2})^n}{(q;q)_n} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^{\frac{ak}{2}(k+n)}}{(q^{n+1};q)_k} \dfrac{(\frac{x^2}{4})^k}{(q;q)_k}.$$

Properties

Theorem: The following formula holds for all $n \in \mathbb{Z}$: $$J_{-n}(x,a;q)=(-1)^{n(a+1)}J_n(x,a;q).$$

Proof: █

Theorem: The following formula holds for $n \in \mathbb{Z}$: $$J_n(-x,a;q)=(-1)^nJ_n(x,a;q).$$

Proof: █

Theorem (Generating function): The following formula holds: $$E_q \left(\dfrac{q^{\frac{a}{4}}xt}{2}, \dfrac{a}{2} \right) E_q \left( \dfrac{(-1)^{a+1}q^{\frac{a}{4}}x}{2t},\dfrac{a}{2} \right) = \displaystyle\sum_{k=-\infty}^{\infty} q^{\frac{ak^2}{4}} J_k(x,a;q)t^k.$$

Proof: █

Theorem: The following formula holds: $$J_n(x,a;q)=\dfrac{2}{x}(1-q^{n+1})J_{n+1}\left(q^{-\frac{a}{4}}x,a;q \right)+(-1)^{a+1}q^{\frac{(a+2)(n+1)}{2}} J_{n+2}(x,a;q).$$

Proof: █