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Statements
V
(
n
+
1
2
,
z
)
=
2
/
π
e
1
4
z
2
(
-
i
)
n
𝐻𝑒
n
(
i
z
)
=
2
/
π
e
1
4
z
2
(
-
i
)
n
2
-
1
2
n
H
n
(
i
z
/
2
)
,
parabolic-V
𝑛
1
2
𝑧
2
𝜋
superscript
𝑒
1
4
superscript
𝑧
2
superscript
𝑖
𝑛
Hermite-polynomial-He
𝑛
𝑖
𝑧
2
𝜋
superscript
𝑒
1
4
superscript
𝑧
2
superscript
𝑖
𝑛
superscript
2
1
2
𝑛
Hermite-polynomial-H
𝑛
𝑖
𝑧
2
{\displaystyle{\displaystyle V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{%
\frac{1}{4}z^{2}}(-i)^{n}\mathit{He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1%
}{4}z^{2}}(-i)^{n}2^{-\frac{1}{2}n}H_{n}\left(iz/\sqrt{2}\right),}}
n
=
0
,
1
,
2
,
…
𝑛
0
1
2
…
{\displaystyle{\displaystyle n=0,1,2,\dots}}
H
n
(
x
)
Hermite-polynomial-H
𝑛
𝑥
{\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}
𝐻𝑒
n
(
x
)
Hermite-polynomial-He
𝑛
𝑥
{\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
V
(
a
,
z
)
parabolic-V
𝑎
𝑧
{\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}
z
𝑧
{\displaystyle{\displaystyle z}}
n
𝑛
{\displaystyle{\displaystyle n}}
