q-Meixner–Pollaczek polynomials

In mathematics, the q-Meixner–Pollaczek 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]

$P_{n}(x;a|q)=a^{-n}e^{in\phi}$$\frac{a^2;q_n}{(q;q)_n}$$_3\Phi_2(q^-n,ae^{i(\theta+2\phi)},ae^{-i\theta};a^2,0|q;q)$

$\displaystyle$

References

1. Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p460,Springer

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248