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Statements
𝐻𝑒
n
(
x
)
=
2
1
2
n
U
(
-
1
2
n
,
1
2
,
1
2
x
2
)
=
2
1
2
(
n
-
1
)
x
U
(
-
1
2
n
+
1
2
,
3
2
,
1
2
x
2
)
=
e
1
4
x
2
U
(
-
n
-
1
2
,
x
)
.
Hermite-polynomial-He
𝑛
𝑥
superscript
2
1
2
𝑛
Kummer-confluent-hypergeometric-U
1
2
𝑛
1
2
1
2
superscript
𝑥
2
superscript
2
1
2
𝑛
1
𝑥
Kummer-confluent-hypergeometric-U
1
2
𝑛
1
2
3
2
1
2
superscript
𝑥
2
superscript
𝑒
1
4
superscript
𝑥
2
parabolic-U
𝑛
1
2
𝑥
{\displaystyle{\displaystyle\mathit{He}_{n}\left(x\right)=2^{\frac{1}{2}n}U%
\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x^{2}\right)=2^{\frac{1}{2}(n-1)%
}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{%
\tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1}{2},x\right).}}
𝐻𝑒
n
(
x
)
Hermite-polynomial-He
𝑛
𝑥
{\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}
U
(
a
,
b
,
z
)
Kummer-confluent-hypergeometric-U
𝑎
𝑏
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
U
(
a
,
z
)
parabolic-U
𝑎
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
