In mathematics, the big 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
P
n
(
x
;
a
,
b
;
q
)
=
1
(
b
−
1
q
−
n
;
q
)
n
2
ϕ
1
(
q
−
n
,
a
q
x
−
1
;
a
q
;
q
,
x
b
)
{\displaystyle P_{n}(x;a,b;q)={\frac {1}{(b^{-1}q^{-n};q)_{n}}}{}_{2}\phi _{1}\left(q^{-n},aqx^{-1};aq;q,{\frac {x}{b}}\right)}
Relation to other polynomials
Big q-Laguerre polynomials→Laguerre polynomials
References
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 