# q-Bessel polynomials

In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

## Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by [1]

${\displaystyle y_{n}(x;a;q)=\;_{2}\phi _{1}\left({\begin{matrix}q^{-N}&-aq^{n}\\0\end{matrix}};q,qx\right)}$

## Orthogonality

${\displaystyle \sum _{k=0}^{\infty }\left({\frac {a^{k}}{(q;q)_{n}}}*q^{k+1 \choose 2}*y_{m}*(q^{k};a;q)*y_{n}*(q^{k};a;q)\right)=(q;q)_{n}*(-aq^{n};q)_{\infty }{\frac {a^{n}*q^{n+1 \choose 2}}{1+aq^{2n}}}\delta _{mn}}$[2]

## Gallery

 QBessel function abs complex 3D Maple plot QBessel function Im complex 3D Maple plot QBessel function Re complex 3D Maple plot
 QBessel function abs density Maple plot QBessel function Im density Maple plot QBessel function Re density Maple plot

## References

1. ^ Roelof Koekoek, Peter Lesky Rene Swarttouw,Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010
2. ^ Roelof p527