- Source: Affine q-Krawtchouk polynomials
In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. 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
aff
(
q
−
x
;
p
;
N
;
q
)
=
3
ϕ
2
(
q
−
n
,
0
,
q
−
x
p
q
,
q
−
N
;
q
,
q
)
,
n
=
0
,
1
,
2
,
…
,
N
.
{\displaystyle K_{n}^{\text{aff}}(q^{-x};p;N;q)={}_{3}\phi _{2}\left({\begin{matrix}q^{-n},0,q^{-x}\\pq,q^{-N}\end{matrix}};q,q\right),\qquad n=0,1,2,\ldots ,N.}
Relation to other polynomials
affine q-Krawtchouk polynomials → little q-Laguerre polynomials：
lim
a
→
1
=
K
n
aff
(
q
x
−
N
;
p
,
N
∣
q
)
=
p
n
(
q
x
;
p
,
q
)
{\displaystyle \lim _{a\to 1}=K_{n}^{\text{aff}}(q^{x-N};p,N\mid q)=p_{n}(q^{x};p,q)}
.
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), "Affine q-Krawtchouk 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.
Stanton, Dennis (1981), "Three addition theorems for some q-Krawtchouk polynomials", Geometriae Dedicata, 10 (1): 403–425, doi:10.1007/BF01447435, ISSN 0046-5755, MR 0608153, S2CID 119838893
Kata Kunci Pencarian:
- Affine q-Krawtchouk polynomials
- Q-Krawtchouk polynomials
- Orthogonal polynomials
- List of q-analogs
- Askey scheme
