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]
P
n
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α
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=
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q
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q
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n
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q
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q
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n
{\displaystyle P_{n}^{(\alpha )}(x|q)={\frac {(q^{\alpha +1};q)_{n}}{(q;q)_{n}}}}
3
ϕ
2
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q
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n
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q
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/
2
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/
4
e
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q
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4
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q
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q
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{\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)}
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, 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 ISBN 978-0-521-19225-5 MR 2723248 
