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 q-Pochhammer symbol by ：
P n ( x ; a ∣ q ) = a − n e i n ϕ ( a 2 ; q ) n ( q ; q ) n
3 ϕ 2 ( q − n , a e i ( θ + 2 ϕ ) , a e − i θ ; a 2 , 0 ∣ q ; q ) , x = cos ( θ + ϕ ) . {\displaystyle P_{n}(x;a\mid 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\mid q;q),\quad x=\cos(\theta +\phi ).}
References
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.
