q-Bessel polynomials

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

Definition

The polynomials are given in terms of basic hypergeometric functions by ^[1]：

y_{n} (x; a; q) =_{2} ϕ_{1} (\begin{matrix} q^{- n} & - a q^{n} \\ 0 \end{matrix}; q, q x) .

Also known as alternative q-Charlier polynomials $K (x; a; q) .$

\sum_{k = 0}^{\infty} (\frac{a^{k}}{(q; q)_{n}} * q^{(\binom{k + 1}{2})} * y_{m} * (q^{k}; a; q) * y_{n} * (q^{k}; a; q)) = (q; q)_{n} * (- a q^{n}; q)_{\infty} \frac{a^{n} * q^{(\binom{n + 1}{2})}}{1 + a q^{2 n}} δ_{m n}

^[2]

where $(q; q)_{n} and (- a q^{n}; q)_{\infty}$ are q-Pochhammer symbols.

↑ Roelof Koekoek, Peter Lesky Rene Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010
↑ Roelof p527