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Statements
h
n
(
sinh
t
|
q
)
=
∑
ℓ
=
0
n
q
1
2
ℓ
(
ℓ
+
1
)
(
q
-
n
;
q
)
ℓ
(
q
;
q
)
ℓ
e
(
n
-
2
ℓ
)
t
=
e
n
t
ϕ
1
1
(
q
-
n
0
;
q
,
-
q
e
-
2
t
)
=
i
-
n
H
n
(
i
sinh
t
|
q
-
1
)
.
continuous-q-inverse-Hermite-polynomial-h
𝑛
𝑡
𝑞
superscript
subscript
ℓ
0
𝑛
superscript
𝑞
1
2
ℓ
ℓ
1
q-Pochhammer-symbol
superscript
𝑞
𝑛
𝑞
ℓ
q-Pochhammer-symbol
𝑞
𝑞
ℓ
superscript
𝑒
𝑛
2
ℓ
𝑡
superscript
𝑒
𝑛
𝑡
q-hypergeometric-rphis
1
1
superscript
𝑞
𝑛
0
𝑞
𝑞
superscript
𝑒
2
𝑡
imaginary-unit
𝑛
continuous-q-Hermite-polynomial-H
𝑛
imaginary-unit
𝑡
superscript
𝑞
1
{\displaystyle{\displaystyle h_{n}\left(\sinh t\,|\,q\right)=\sum_{\ell=0}^{n}%
q^{\frac{1}{2}\ell(\ell+1)}\frac{\left(q^{-n};q\right)_{\ell}}{\left(q;q\right%
)_{\ell}}e^{(n-2\ell)t}=e^{nt}{{}_{1}\phi_{1}}\left({q^{-n}\atop 0};q,-qe^{-2t%
}\right)={\mathrm{i}^{-n}}H_{n}\left(\mathrm{i}\sinh t\,|\,q^{-1}\right).}}
H
n
(
x
|
q
)
continuous-q-Hermite-polynomial-H
𝑛
𝑥
𝑞
{\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\,|\,\NVar{q}\right)}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
(
a
;
q
)
n
q-Pochhammer-symbol
𝑎
𝑞
𝑛
{\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
q-hypergeometric-rphis
𝑟
1
𝑠
subscript
𝑎
0
…
subscript
𝑎
𝑟
subscript
𝑏
1
…
subscript
𝑏
𝑠
𝑞
𝑧
{\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},%
\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
q
𝑞
{\displaystyle{\displaystyle q}}
ℓ
ℓ
{\displaystyle{\displaystyle\ell}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
