From Wikipedia, the free encyclopedia
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.
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
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 , Cambridge University Press,
ISBN
978-0-521-19225-5 ,
MR
2723248 .