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 by ^[1]：

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

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

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]

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

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

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.