Q-Meixner–Pollaczek polynomials
In mathematics, the q -Meixner–Pollaczek polynomialsorthogonal 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 ：[ 1]
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
^ Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p 460, Springer
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 ISBN 978-0-521-19225-5 MR 2723248
