（en:Incomplete gamma function - oldid=701538344 ）
第二種不完全ガンマ関数
Γ
(
s
,
x
)
{\displaystyle \Gamma (s,x)}
x
{\displaystyle x}
∂
∂
x
Γ
(
s
,
x
)
=
−
x
s
−
1
e
−
x
{\displaystyle {\frac {\partial }{\partial x}}\Gamma (s,x)=-x^{s-1}e^{-x}}
s
{\displaystyle s}
[ 1]
∂
∂
s
Γ
(
s
,
x
)
=
(
ln
x
)
Γ
(
s
,
x
)
+
x
T
(
3
,
s
,
x
)
{\displaystyle {\frac {\partial }{\partial s}}\Gamma (s,x)=(\ln x)\Gamma (s,x)+x\,T(3,s,x)}
二階微分は次のようになる。
∂
2
∂
s
2
Γ
(
s
,
x
)
=
(
ln
x
)
2
Γ
(
s
,
x
)
+
2
x
[
(
ln
x
)
T
(
3
,
s
,
x
)
+
T
(
4
,
s
,
x
)
]
{\displaystyle {\frac {\partial ^{2}}{\partial s^{2}}}\Gamma (s,x)=(\ln x)^{2}\Gamma (s,x)+2x{\bigl [}(\ln x)\,T(3,s,x)+T(4,s,x){\bigr ]}}
ここで、関数
T
(
m
,
s
,
x
)
{\displaystyle T(m,s,x)}
マイヤーのG関数 （英語版 ）
T
(
m
,
s
,
x
)
=
G
m
−
1
,
m
m
,
0
(
0
,
0
,
…
,
0
s
−
1
,
−
1
,
…
,
−
1
|
x
)
.
{\displaystyle T(m,s,x)=G_{m-1,\,m}^{\,m,\,0}\!\left(\left.{\begin{matrix}0,0,\dots ,0\\s-1,-1,\dots ,-1\end{matrix}}\;\right|\,x\right).}
この
T
{\displaystyle T}
n
{\displaystyle n}
この
T
{\displaystyle T}
[訳語疑問点
∂
m
∂
s
m
Γ
(
s
,
x
)
=
ln
m
x
Γ
(
s
,
x
)
+
m
x
∑
n
=
0
m
−
1
(
m
−
1
)
n
ln
m
−
n
−
1
x
T
(
3
+
n
,
s
,
x
)
{\displaystyle {\frac {\partial ^{m}}{\partial s^{m}}}\Gamma (s,x)=\ln ^{m}x\Gamma (s,x)+mx\,\sum _{n=0}^{m-1}(m-1)_{n}\ln ^{m-n-1}x\,T(3+n,s,x)}
ここで、
(
n
)
i
{\displaystyle (n)_{i}}
階乗冪 （ポッホハマー記号 ）を表す。
これらの導関数は、次の式によって次々と求めることができる。
∂
∂
s
T
(
m
,
s
,
x
)
=
ln
x
T
(
m
,
s
,
x
)
+
(
m
−
1
)
T
(
m
+
1
,
s
,
x
)
∂
∂
x
T
(
m
,
s
,
x
)
=
−
1
x
[
T
(
m
−
1
,
s
,
x
)
+
T
(
m
,
s
,
x
)
]
{\displaystyle {\begin{aligned}{\frac {\partial }{\partial s}}T(m,s,x)&=\ln x~T(m,s,x)+(m-1)T(m+1,s,x)\\{\frac {\partial }{\partial x}}T(m,s,x)&=-{\frac {1}{x}}[T(m-1,s,x)+T(m,s,x)]\end{aligned}}}
T
(
m
,
s
,
z
)
{\displaystyle T(m,s,z)}
|
z
|
<
1
{\displaystyle |z|<1}
T
(
m
,
s
,
z
)
=
−
(
−
1
)
m
−
1
(
m
−
2
)
!
d
m
−
2
d
t
m
−
2
[
Γ
(
s
−
t
)
z
t
−
1
]
|
t
=
0
+
∑
n
=
0
∞
(
−
1
)
n
z
s
−
1
+
n
n
!
(
−
s
−
n
)
m
−
1
{\displaystyle T(m,s,z)=-{\frac {(-1)^{m-1}}{(m-2)!}}{\frac {{\rm {d}}^{m-2}}{{\rm {d}}t^{m-2}}}\left[\Gamma (s-t)z^{t-1}\right]{\Big |}_{t=0}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{s-1+n}}{n!(-s-n)^{m-1}}}}
^ K.O. Geddes , M.L. Glasser, R.A. Moore and T.C. Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions , AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149-165, [1]
![{\displaystyle {\frac {\partial ^{2}}{\partial s^{2}}}\Gamma (s,x)=(\ln x)^{2}\Gamma (s,x)+2x{\bigl [}(\ln x)\,T(3,s,x)+T(4,s,x){\bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fdcd122178e4500baeb5d0cf64a49bfc934065)
![{\displaystyle {\begin{aligned}{\frac {\partial }{\partial s}}T(m,s,x)&=\ln x~T(m,s,x)+(m-1)T(m+1,s,x)\\{\frac {\partial }{\partial x}}T(m,s,x)&=-{\frac {1}{x}}[T(m-1,s,x)+T(m,s,x)]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf82022e83f730cdabb7987624ce744a10adb068)
![{\displaystyle T(m,s,z)=-{\frac {(-1)^{m-1}}{(m-2)!}}{\frac {{\rm {d}}^{m-2}}{{\rm {d}}t^{m-2}}}\left[\Gamma (s-t)z^{t-1}\right]{\Big |}_{t=0}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{s-1+n}}{n!(-s-n)^{m-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9d7c30bc1fcd8805304d0556ca97cef8b4e049)