F
(
x
)
{\displaystyle F(x)\,}
f
H
(
n
)
{\displaystyle f_{H}(n)\,}
x
m
{\displaystyle x^{m}}
{
m
!
π
2
m
−
n
(
m
−
n
2
)
!
,
(
m
−
n
)
even and
≥
0
0
,
otherwise
{\displaystyle {\begin{cases}{\frac {m!{\sqrt {\pi }}}{2^{m-n}\left({\frac {m-n}{2}}\right)!}},&(m-n){\text{ even and}}\geq 0\\0,&{\text{otherwise}}\end{cases}}}
[ 1]
e
a
x
{\displaystyle e^{ax}\,}
π
a
n
e
a
2
/
4
{\displaystyle {\sqrt {\pi }}a^{n}e^{a^{2}/4}\,}
e
2
x
t
−
t
2
,
|
t
|
<
1
2
{\displaystyle e^{2xt-t^{2}},\ |t|<{\frac {1}{2}}\,}
π
(
2
t
)
n
{\displaystyle {\sqrt {\pi }}(2t)^{n}}
H
m
(
x
)
{\displaystyle H_{m}(x)\,}
π
2
n
n
!
δ
n
m
{\displaystyle {\sqrt {\pi }}2^{n}n!\delta _{nm}\,}
x
2
H
m
(
x
)
{\displaystyle x^{2}H_{m}(x)\,}
2
n
n
!
π
{
1
,
n
=
m
+
2
(
n
+
1
2
)
,
n
=
m
(
n
+
1
)
(
n
+
2
)
,
n
=
m
−
2
0
,
otherwise
{\displaystyle 2^{n}n!{\sqrt {\pi }}{\begin{cases}1,&n=m+2\\\left(n+{\frac {1}{2}}\right),&n=m\\(n+1)(n+2),&n=m-2\\0,&{\text{otherwise}}\end{cases}}}
e
−
x
2
H
m
(
x
)
{\displaystyle e^{-x^{2}}H_{m}(x)\,}
(
−
1
)
p
−
m
2
p
−
1
/
2
Γ
(
p
+
1
/
2
)
,
m
+
n
=
2
p
,
p
∈
Z
{\displaystyle \left(-1\right)^{p-m}2^{p-1/2}\Gamma (p+1/2),\ m+n=2p,\ p\in \mathbb {Z} }
H
m
2
(
x
)
{\displaystyle H_{m}^{2}(x)\,}
{
2
m
+
n
/
2
π
(
m
n
/
2
)
m
!
n
!
(
n
/
2
)
!
,
n
even and
≤
2
m
0
,
otherwise
{\displaystyle {\begin{cases}2^{m+n/2}{\sqrt {\pi }}{\binom {m}{n/2}}{\frac {m!n!}{(n/2)!}},&n{\text{ even and}}\leq 2m\\0,&{\text{otherwise}}\end{cases}}}
[ 2]
H
m
(
x
)
H
p
(
x
)
{\displaystyle H_{m}(x)H_{p}(x)\,}
{
2
k
π
m
!
n
!
p
!
(
k
−
m
)
!
(
k
−
n
)
!
(
k
−
p
)
!
,
n
+
m
+
p
=
2
k
,
k
∈
Z
;
|
m
−
p
|
≤
n
≤
m
+
p
0
,
otherwise
{\displaystyle {\begin{cases}{\frac {2^{k}{\sqrt {\pi }}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&n+m+p=2k,\ k\in \mathbb {Z} ;\ |m-p|\leq n\leq m+p\\0,&{\text{otherwise}}\end{cases}}\,}
[ 3]
H
n
+
p
+
q
(
x
)
H
p
(
x
)
H
q
(
x
)
{\displaystyle H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,}
π
2
n
+
p
+
q
(
n
+
p
+
q
)
!
{\displaystyle {\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,}
d
m
d
x
m
F
(
x
)
{\displaystyle {\frac {d^{m}}{dx^{m}}}F(x)\,}
f
H
(
n
+
m
)
{\displaystyle f_{H}(n+m)\,}
x
d
m
d
x
m
F
(
x
)
{\displaystyle x{\frac {d^{m}}{dx^{m}}}F(x)\,}
n
f
H
(
n
+
m
−
1
)
+
1
2
f
H
(
n
+
m
+
1
)
{\displaystyle nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,}
e
x
2
d
d
x
[
e
−
x
2
d
d
x
F
(
x
)
]
{\displaystyle e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,}
−
2
n
f
H
(
n
)
{\displaystyle -2nf_{H}(n)\,}
F
(
x
−
x
0
)
{\displaystyle F(x-x_{0})}
π
∑
k
=
0
∞
(
−
x
0
)
k
k
!
f
H
(
n
+
k
)
{\displaystyle {\sqrt {\pi }}\sum _{k=0}^{\infty }{\frac {(-x_{0})^{k}}{k!}}f_{H}(n+k)}
F
(
x
)
∗
G
(
x
)
{\displaystyle F(x)*G(x)\,}
π
(
−
1
)
n
[
2
2
n
+
1
Γ
(
n
+
3
2
)
]
−
1
f
H
(
n
)
g
H
(
n
)
{\displaystyle {\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,}
[ 4]
e
z
2
sin
(
x
z
)
,
|
z
|
<
1
2
{\displaystyle e^{z^{2}}\sin(xz),\ |z|<{\frac {1}{2}}\ \,}
{
π
(
−
1
)
⌊
n
2
⌋
(
2
z
)
n
,
n
o
d
d
0
,
n
e
v
e
n
{\displaystyle {\begin{cases}{\sqrt {\pi }}(-1)^{\lfloor {\frac {n}{2}}\rfloor }(2z)^{n},&n\,\mathrm {odd} \\0,&n\,\mathrm {even} \end{cases}}\,}
(
1
−
z
2
)
−
1
/
2
exp
[
2
x
y
z
−
(
x
2
+
y
2
)
z
2
(
1
−
z
2
)
]
{\displaystyle (1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,}
π
z
n
H
n
(
y
)
{\displaystyle {\sqrt {\pi }}z^{n}H_{n}(y)}
[ 5] [ 6]
H
m
(
y
)
H
m
+
1
(
x
)
−
H
m
(
x
)
H
m
+
1
(
y
)
2
m
+
1
m
!
(
x
−
y
)
{\displaystyle {\frac {H_{m}(y)H_{m+1}(x)-H_{m}(x)H_{m+1}(y)}{2^{m+1}m!(x-y)}}}
{
π
H
n
(
y
)
n
≤
m
0
n
>
m
{\displaystyle {\begin{cases}{\sqrt {\pi }}H_{n}(y)&n\leq m\\0&n>m\end{cases}}}
![{\displaystyle e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c7d97bfe8587a9904ba8332c64f693b13eba3c)
![{\displaystyle {\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81c4c392666d42cce6c9dc6ea75b4c88215bfebc)
![{\displaystyle (1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/103e6fef22d2baa4b64de7f7956c46e8ff9ede7f)
