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 Pochhammer symbol by

$\displaystyle$

Orthogonality

Recurrence and difference relations

Rodrigues formula

Generating function

Relation to other polynomials

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, ISBN 978-0-8218-2321-7, ISSN 0065-9266, MR783216, http://books.google.com/books?id=9q9o03nD_xsC 
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR2128719 
• 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, MR2656096 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/18 
• 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, MR634153