q-Meixner–Pollaczek polynomials
From HandWiki
In mathematics, the q-Meixner–Pollaczek polynomials 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 ：[1]
- [math]\displaystyle{ P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{(a^2;q)_n}{(q;q)_n}{}_3\phi_2(q^{-n}, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q),\quad x=\cos(\theta+\phi). }[/math]
References
- ↑ Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analoques, p 460, Springer
- 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
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