From Wikipedia, the free encyclopedia
In mathematics, the q -Racah polynomials are a family of basic hypergeometric
orthogonal polynomials in the basic
Askey scheme , introduced by
Askey & Wilson (1979) . Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (
2010 , 14) give a detailed list of their properties.
The polynomials are given in terms of
basic hypergeometric functions and the
Pochhammer symbol by
p
n
(
q
−
x
+
q
x
+
1
c
d
;
a
,
b
,
c
,
d
;
q
)
=
4
ϕ
3
q
−
n
a
b
q
n
+
1
q
−
x
q
x
+
1
c
d
a
q
b
d
q
c
q
;
q
;
q
{\displaystyle p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right]}
They are sometimes given with changes of variables as
W
n
(
x
;
a
,
b
,
c
,
N
;
q
)
=
4
ϕ
3
q
−
n
a
b
q
n
+
1
q
−
x
c
q
x
−
n
a
q
b
c
q
q
−
N
;
q
;
q
{\displaystyle W_{n}(x;a,b,c,N;q)={}_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&cq^{x-n}\\aq&bcq&q^{-N}\\\end{matrix}};q;q\right]}
Relation to other polynomials
q-Racah polynomials→Racah polynomials
Askey, Richard; Wilson, James (1979),
"A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols" , SIAM Journal on Mathematical Analysis , 10 (5): 1008–1016,
doi :
10.1137/0510092 ,
ISSN
0036-1410 ,
MR
0541097 , archived from
the original on September 25, 2017
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 .