Continuous q-Laguerre polynomials
From HandWiki
In mathematics, the continuous q-Laguerre 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}^{(\alpha)}(x|q)=\frac{(q^{\alpha+1};q)_{n}}{(q;q)_{n}} }[/math][math]\displaystyle{ _{3}\phi_{2}(q^{-n},q^{\alpha/2+1/4}e^{i\theta},q^{\alpha/2+1/4}e^{-i\theta};q^{\alpha+1},0|q,q) }[/math]
References
- ↑ Roelof Koekoek, Peter Lesky, Rene Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, p514, 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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