Q-Bessel polynomials

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition
The polynomials are given in terms of basic hypergeometric functions by ：


 * $$y_{n}(x;a;q)=\;{}_2\phi_1 \left(\begin{matrix}

q^{-n} & -aq^{n} \\ 0 \end{matrix} Also known as alternative q-Charlier polynomials $$K(x;a;q).$$
 * q,qx \right). $$

Orthogonality


\sum_{k=0}^{\infty}\left(\frac{a^k}{(q;q)_n}*q^{k+1 \choose 2}*y_{m}*(q^k;a;q)*y_{n}*(q^k;a;q)\right)=(q;q)_{n}*(-aq^n;q)_{\infty}\frac{ a^{n}*q^{n+1 \choose 2} }{1+aq^{2n}}\delta_{mn} $$ where $$(q;q)_n\text{ and }(-aq^n;q)_\infty$$ are q-Pochhammer symbols.