Continuous q-Jacobi 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

$P_{n}^{(\alpha ,\beta )}(x;q)={\frac {(q^{n+1};q)_{n}}{(q;q)_{n}}}{}_{4}\phi _{3}\left[{\begin{matrix}q^{-n},q^{n+\alpha +\beta +1},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{i\theta }},q^{{\frac {1}{2}}\alpha +{\frac {1}{4}}e^{-i\theta }}\\q^{n+1},-q^{{\frac {1}{2}}(\alpha +\beta +1)},-q^{{\frac {1}{2}}(\alpha +\beta +2)}\end{matrix}};q,q\right]\qquad x=\cos \,\theta .$

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.