Continuous q-Jacobi polynomials
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 α -->
,
β β -->
)
(
x
;
q
)
=
(
q
n
+
1
;
q
)
n
(
q
;
q
)
n
4
ϕ ϕ -->
3
[
q
− − -->
n
,
q
n
+
α α -->
+
β β -->
+
1
,
q
1
2
α α -->
+
1
4
e
i
θ θ -->
,
q
1
2
α α -->
+
1
4
e
− − -->
i
θ θ -->
q
n
+
1
,
− − -->
q
1
2
(
α α -->
+
β β -->
+
1
)
,
− − -->
q
1
2
(
α α -->
+
β β -->
+
2
)
;
q
,
q
]
x
=
cos
θ 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 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
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd18c7ed289efd168269972da5d18b89ed13bde)
