q-Charlier polynomials

In mathematics, the q-Charlier polynomials^[1] 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 the basic hypergeometric function by

\displaystyle C_{n}(q^{-x};a;q)={}_{2}\phi _{1}(q^{-n},q^{-x};0;q,-q^{n+1}/a).

References

^ There are similar named polynomials named alternative q-Charlier polynomials $K_{n}(x;a;q)$ which is another name for q-Bessel polynomials.

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), "Chapter 18: 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.
Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.

[1] There are similar named polynomials named alternative q-Charlier polynomials $K_{n}(x;a;q)$ which is another name for q-Bessel polynomials.

[1]