Ein
q
{\displaystyle q}
q
{\displaystyle q}
Kombinatorik auftritt. Ein
q
{\displaystyle q}
q
{\displaystyle q}
q
→
1
{\displaystyle q\to 1}
speziellen Funktionen insbesondere in der Theorie der
q
{\displaystyle q}
Eine natürliche Zahl
n
{\displaystyle n}
q
{\displaystyle q}
[
n
]
q
:=
1
−
q
n
1
−
q
=
1
+
q
+
q
2
+
⋯
+
q
n
−
1
,
{\displaystyle [n]_{q}:={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\cdots +q^{n-1},}
da
lim
q
→
1
[
n
]
q
=
n
{\displaystyle \lim _{q\rightarrow 1}[n]_{q}=n}
Die
q
{\displaystyle q}
n
>
0
{\displaystyle n>0}
[1]
[
n
]
q
!
:=
[
n
]
q
[
n
−
1
]
q
⋯
[
1
]
q
=
∏
k
=
1
n
1
−
q
k
1
−
q
,
{\displaystyle [n]_{q}!:=[n]_{q}[n-1]_{q}\cdots [1]_{q}=\prod \limits _{k=1}^{n}{\frac {1-q^{k}}{1-q}},}
und
[
0
]
q
!
:=
1
{\displaystyle [0]_{q}!:=1}
Durch ausmultiplizieren erhält man
[
n
]
q
!
=
1
⋅
(
1
+
q
)
⋯
(
1
+
q
+
⋯
+
q
n
−
2
)
⋅
(
1
+
q
+
⋯
+
q
n
−
1
)
.
{\displaystyle [n]_{q}!=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1}).}
Das
q
{\displaystyle q}
q
{\displaystyle q}
(
a
;
q
)
n
:=
∏
k
=
1
n
(
1
−
a
q
k
−
1
)
{\displaystyle (a;q)_{n}:=\prod \limits _{k=1}^{n}(1-aq^{k-1})}
oder allgemeiner
(
a
1
,
a
2
,
…
,
a
m
;
q
)
n
:=
(
a
1
;
q
)
n
(
a
2
;
q
)
n
…
(
a
m
;
q
)
n
.
{\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}:=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.}
Der
q
{\displaystyle q}
(
n
k
)
q
:=
[
n
]
q
!
[
k
]
q
!
[
n
−
k
]
q
!
=
∏
j
=
1
k
(
1
−
q
n
−
j
+
1
)
(
1
−
q
j
)
.
{\displaystyle {\binom {n}{k}}_{q}:={\frac {[n]_{q}!}{[k]_{q}![n-k]_{q}!}}=\prod \limits _{j=1}^{k}{\frac {(1-q^{n-j+1})}{(1-q^{j})}}.}
Es gilt
[
n
]
q
!
=
(
q
;
q
)
n
(
1
−
q
)
n
{\displaystyle [n]_{q}!={\frac {(q;q)_{n}}{(1-q)^{n}}}}
und
(
n
k
)
q
=
(
q
;
q
)
n
(
q
;
q
)
k
(
q
;
q
)
n
−
k
.
{\displaystyle {\binom {n}{k}}_{q}={\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}.}
Das
q
{\displaystyle q}
verallgemeinerten hypergeometrischen Funktion ist die
q
{\displaystyle q}
[1]
r
ϕ
s
[
a
1
a
2
…
a
r
b
1
b
2
…
b
s
;
q
,
z
]
=
∑
n
=
0
∞
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
(
q
,
b
1
,
b
2
,
…
,
b
s
;
q
)
n
z
n
(
−
q
(
n
−
1
)
/
2
)
n
(
s
+
1
−
r
)
.
{\displaystyle \;_{r}\phi _{s}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{r}\\b_{1}&b_{2}&\ldots &b_{s}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{r};q)_{n}}{(q,b_{1},b_{2},\ldots ,b_{s};q)_{n}}}z^{n}\left(-q^{(n-1)/2}\right)^{n(s+1-r)}.}
Die stetigen
q
{\displaystyle q}
{
H
n
(
x
∣
q
)
}
{\displaystyle \{H_{n}(x\mid q)\}}
[2]
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)}
mit Anfangswerten
H
0
(
x
∣
q
)
=
1
,
H
1
(
x
∣
q
)
=
2
x
.
{\displaystyle H_{0}(x\mid q)=1,H_{1}(x\mid q)=2x.}
Das
q
{\displaystyle q}
Exponentialfunktion ist
e
q
x
:=
∑
n
=
0
∞
x
n
[
n
]
q
!
.
{\displaystyle e_{q}^{x}:=\sum \limits _{n=0}^{\infty }{\frac {x^{n}}{[n]_{q}!}}.}
Das
q
{\displaystyle q}
Ableitung einer Funktion
f
{\displaystyle f}
Q-Differenz
(
D
q
f
)
(
x
)
=
f
(
x
)
−
f
(
q
x
)
(
1
−
q
)
x
,
{\displaystyle (D_{q}f)(x)={\frac {f(x)-f(qx)}{(1-q)x}},}
dadurch entsteht das sogenannte
q
{\displaystyle q}
Das
q
{\displaystyle q}
(
x
−
a
)
n
{\displaystyle (x-a)^{n}}
(
x
−
a
)
q
n
:=
∏
k
=
0
n
−
1
(
x
−
q
k
a
)
,
{\displaystyle (x-a)_{q}^{n}:=\prod \limits _{k=0}^{n-1}(x-q^{k}a),}
zusammen mit der
q
{\displaystyle q}
q
{\displaystyle q}
q
{\displaystyle q}
Taylorreihe für
f
{\displaystyle f}
f
(
x
)
=
∑
n
=
0
∞
D
q
n
f
(
a
)
(
x
−
a
)
q
n
[
n
]
q
!
.
{\displaystyle f(x)=\sum \limits _{n=0}^{\infty }{\frac {D_{q}^{n}f(a)(x-a)_{q}^{n}}{[n]_{q}!}}.}
Mourad E.H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable . Hrsg.: Cambridge University Press. 2005, doi :10.1017/CBO9781107325982
↑ a b Mourad E.H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable . Hrsg.: Cambridge University Press. 2005, S. 299 , doi :10.1017/CBO9781107325982
↑ Mourad E.H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable . Hrsg.: Cambridge University Press. 2005, S. 319 , doi :10.1017/CBO9781107325982
![{\displaystyle [n]_{q}:={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\cdots +q^{n-1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1229d48f813c876ff707ad8166e1124d42311f44)
![{\displaystyle \lim _{q\rightarrow 1}[n]_{q}=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7aa0e24abc7002668ed0dfb715146b12667c1cd)
![{\displaystyle [n]_{q}!:=[n]_{q}[n-1]_{q}\cdots [1]_{q}=\prod \limits _{k=1}^{n}{\frac {1-q^{k}}{1-q}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2df14de70919ff373013473c25be7314ac4c94f0)
![{\displaystyle [0]_{q}!:=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5fe7802f3a639d11cd85f58a1d82e7bf38dd19)
![{\displaystyle [n]_{q}!=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8f485fa026fd2038b3ec8cf0d9e87e3d40c4d7)
![{\displaystyle {\binom {n}{k}}_{q}:={\frac {[n]_{q}!}{[k]_{q}![n-k]_{q}!}}=\prod \limits _{j=1}^{k}{\frac {(1-q^{n-j+1})}{(1-q^{j})}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acf9a19e087b5154a568627a24edf726cc23290e)
![{\displaystyle [n]_{q}!={\frac {(q;q)_{n}}{(1-q)^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44a4a802c46c694da369a6ec8eb99270e1a0ff9b)
![{\displaystyle \;_{r}\phi _{s}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{r}\\b_{1}&b_{2}&\ldots &b_{s}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{r};q)_{n}}{(q,b_{1},b_{2},\ldots ,b_{s};q)_{n}}}z^{n}\left(-q^{(n-1)/2}\right)^{n(s+1-r)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee6cb2d96fb9effb761102d5e9107862f3da590)
![{\displaystyle e_{q}^{x}:=\sum \limits _{n=0}^{\infty }{\frac {x^{n}}{[n]_{q}!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e67725d2471da70f47286cf5c9ffbce92d0d2d)
![{\displaystyle f(x)=\sum \limits _{n=0}^{\infty }{\frac {D_{q}^{n}f(a)(x-a)_{q}^{n}}{[n]_{q}!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2e2a813996d5e4dc4085e03738ee561a753721)