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.

The polynomials are given in terms of the basic hypergeometric function by

C_{n} (q^{- x}; a; q) =_{2} ϕ_{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, 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
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Sadjang, Patrick Njionou (2013). Moments of Classical Orthogonal Polynomials (Ph.D. thesis). University of Kassel.

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[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]