Continuous q-Jacobi polynomials

Short description: Family of orthogonal polynomials

In mathematics, the continuous q-Jacobi polynomials P(α,β)n(x|q), introduced by (Askey Wilson), 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

[math]\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^{\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. }[/math]

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, https://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, ISBN 978-0-521-83357-8 
• 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 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, 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 
• Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.