Dual q-Krawtchouk polynomials
In mathematics, the dual q -Krawtchouk 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 by
K
n
(
λ
(
x
)
;
c
,
N
|
q
)
=
3
ϕ
2
(
q
−
n
,
q
−
x
,
c
q
x
−
N
;
q
−
N
,
0
|
q
;
q
)
,
n
=
0
,
1
,
2
,
.
.
.
,
N
,
{\displaystyle K_{n}(\lambda (x);c,N|q)={}_{3}\phi _{2}(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q),\quad n=0,1,2,...,N,}
where
λ
(
x
)
=
q
−
x
+
c
q
x
−
N
.
{\displaystyle \lambda (x)=q^{-x}+cq^{x-N}.}
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 ISBN 978-0-521-19225-5 MR 2723248
