Continuous q-Jacobi polynomials
{{Short description|Family of orthogonal polynomials}}
{{DISPLAYTITLE:Continuous q-Jacobi polynomials}}
In mathematics, the continuous q-Jacobi polynomials P{{su|b=n|p=(α,β)}}(x|q), introduced by {{harvtxt|Askey|Wilson|1985}}, are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=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
:
q^{-n},q^{n+\alpha+\beta+1},q^{\frac12\alpha+\frac14e^{i\theta}},q^{\frac12\alpha+\frac14e^{-i\theta}}\\
q^{n+1},-q^{\frac12(\alpha+\beta+1)},-q^{\frac12(\alpha+\beta+2)}\end{matrix}
;q,q\right]\qquad x=\cos\,\theta.
References
- {{Citation | authorlink=Richard Askey | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials | isbn=978-0-8218-2321-7 | mr=783216 | year=1985 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=54 | issue=319 | pages=iv+55|url=https://books.google.com/books?id=9q9o03nD_xsC | doi=10.1090/memo/0319}}
- {{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=Cambridge University Press | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
- {{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
- {{dlmf|id=18|title=Chapter 18: Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
- {{Citation | last1=Rahman | first1=Mizan | title=The linearization of the product of continuous q-Jacobi polynomials | doi=10.4153/CJM-1981-076-8 | mr=634153 | year=1981 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=33 | issue=4 | pages=961–987| s2cid=119464731 | doi-access=free }}
- {{cite thesis |last=Sadjang |first=Patrick Njionou |title=Moments of Classical Orthogonal Polynomials |type=Ph.D. |publisher=Universität Kassel |citeseerx=10.1.1.643.3896 }}