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Family of orthogonal polynomials
In mathematics, the continuous q -Jacobi polynomials P (α,β) n (x |q ), introduced by
Askey & Wilson (1985) , 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
(
α
,
β
)
(
x
;
q
)
=
(
q
n
+
1
;
q
)
n
(
q
;
q
)
n
4
ϕ
3
q
−
n
,
q
n
+
α
+
β
+
1
,
q
1
2
α
+
1
4
e
i
θ
,
q
1
2
α
+
1
4
e
−
i
θ
q
n
+
1
,
−
q
1
2
(
α
+
β
+
1
)
,
−
q
1
2
(
α
+
β
+
2
)
;
q
,
q
x
=
cos
θ
.
{\displaystyle P_{n}^{(\alpha ,\beta )}(x;q)={\frac {(q^{n+1};q)_{n}}{(q;q)_{n}}}{}_{4}\phi _{3}\left[{\begin{matrix}q^{-n},q^{n+\alpha +\beta +1},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{i\theta }},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{-i\theta }}\\q^{n+1},-q^{{\frac {1}{2}}(\alpha +\beta +1)},-q^{{\frac {1}{2}}(\alpha +\beta +2)}\end{matrix}};q,q\right]\qquad x=\cos \,\theta .}
References
Askey, Richard ; Wilson, James (1985),
"Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials" , Memoirs of the American Mathematical Society , 54 (319): iv+55,
doi :
10.1090/memo/0319 ,
ISBN
978-0-8218-2321-7 ,
ISSN
0065-9266 ,
MR
0783216
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 .
Rahman, Mizan (1981), "The linearization of the product of continuous q-Jacobi polynomials",
Canadian Journal of Mathematics , 33 (4): 961–987,
doi :
10.4153/CJM-1981-076-8 ,
ISSN
0008-414X ,
MR
0634153 ,
S2CID
119464731
Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel.
CiteSeerX
10.1.1.643.3896 .