From Wikipedia, the free encyclopedia
In mathematics, the continuous q -Hermite 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 by
H
n
(
x
|
q
)
=
e
i
n
θ
2
ϕ
0
q
−
n
,
0
−
;
q
,
q
n
e
−
2
i
θ
,
x
=
cos
θ
.
{\displaystyle H_{n}(x|q)=e^{in\theta }{}_{2}\phi _{0}\left[{\begin{matrix}q^{-n},0\\-\end{matrix}};q,q^{n}e^{-2i\theta }\right],\quad x=\cos \,\theta .}
Recurrence and difference relations
2
x
H
n
(
x
∣
q
)
=
H
n
+
1
(
x
∣
q
)
+
(
1
−
q
n
)
H
n
−
1
(
x
∣
q
)
{\displaystyle 2xH_{n}(x\mid q)=H_{n+1}(x\mid q)+(1-q^{n})H_{n-1}(x\mid q)}
with the initial conditions
H
0
(
x
∣
q
)
=
1
,
H
−
1
(
x
∣
q
)
=
0
{\displaystyle H_{0}(x\mid q)=1,H_{-1}(x\mid q)=0}
From the above, one can easily calculate:
H
0
(
x
∣
q
)
=
1
H
1
(
x
∣
q
)
=
2
x
H
2
(
x
∣
q
)
=
4
x
2
−
(
1
−
q
)
H
3
(
x
∣
q
)
=
8
x
3
−
2
x
(
2
−
q
−
q
2
)
H
4
(
x
∣
q
)
=
16
x
4
−
4
x
2
(
3
−
q
−
q
2
−
q
3
)
+
(
1
−
q
−
q
3
+
q
4
)
{\displaystyle {\begin{aligned}H_{0}(x\mid q)&=1\\H_{1}(x\mid q)&=2x\\H_{2}(x\mid q)&=4x^{2}-(1-q)\\H_{3}(x\mid q)&=8x^{3}-2x(2-q-q^{2})\\H_{4}(x\mid q)&=16x^{4}-4x^{2}(3-q-q^{2}-q^{3})+(1-q-q^{3}+q^{4})\end{aligned}}}
Generating function
∑
n
=
0
∞
H
n
(
x
∣
q
)
t
n
(
q
;
q
)
n
=
1
(
t
e
i
θ
,
t
e
−
i
θ
;
q
)
∞
{\displaystyle \sum _{n=0}^{\infty }H_{n}(x\mid q){\frac {t^{n}}{(q;q)_{n}}}={\frac {1}{\left(te^{i\theta },te^{-i\theta };q\right)_{\infty }}}}
where
x
=
cos
θ
{\displaystyle \textstyle x=\cos \theta }
.
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 , Cambridge University Press,
ISBN
978-0-521-19225-5 ,
MR
2723248 .
Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel.
CiteSeerX
10.1.1.643.3896 .