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In mathematics , the multivariate gamma function , Γp gamma function . It is useful in multivariate statistics , appearing in the probability density function of the Wishart and inverse Wishart distributions .
It has two equivalent definitions. One is
Γ
p
(
a
)
=
∫
S
>
0
exp
(
−
t
r
a
c
e
(
S
)
)
|
S
|
a
−
(
p
+
1
)
/
2
d
S
,
{\displaystyle \Gamma _{p}(a)=\int _{S>0}\exp \left(-{\rm {trace}}(S)\right)\left|S\right|^{a-(p+1)/2}dS,}
where S>0 means S is positive-definite . The other one, more useful in practice, is
Γ
p
(
a
)
=
π
p
(
p
−
1
)
/
4
∏
j
=
1
p
Γ
[
a
+
(
1
−
j
)
/
2
]
.
{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left[a+(1-j)/2\right].}
From this, we have the recursive relationships:
Γ
p
(
a
)
=
π
(
p
−
1
)
/
2
Γ
(
a
)
Γ
p
−
1
(
a
−
1
2
)
=
π
(
p
−
1
)
/
2
Γ
p
−
1
(
a
)
Γ
[
a
+
(
1
−
p
)
/
2
]
.
{\displaystyle \Gamma _{p}(a)=\pi ^{(p-1)/2}\Gamma (a)\Gamma _{p-1}(a-{\tfrac {1}{2}})=\pi ^{(p-1)/2}\Gamma _{p-1}(a)\Gamma [a+(1-p)/2].}
Thus
Γ
1
(
a
)
=
Γ
(
a
)
{\displaystyle \Gamma _{1}(a)=\Gamma (a)}
Γ
2
(
a
)
=
π
1
/
2
Γ
(
a
)
Γ
(
a
−
1
/
2
)
{\displaystyle \Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)}
Γ
3
(
a
)
=
π
3
/
2
Γ
(
a
)
Γ
(
a
−
1
/
2
)
Γ
(
a
−
1
)
{\displaystyle \Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)}
and so on.
Derivatives
We may define the multivariate digamma function as
ψ
p
(
a
)
=
∂
log
Γ
p
(
a
)
∂
a
=
∑
i
=
1
p
ψ
(
a
+
(
1
−
i
)
/
2
)
,
{\displaystyle \psi _{p}(a)={\frac {\partial \log \Gamma _{p}(a)}{\partial a}}=\sum _{i=1}^{p}\psi (a+(1-i)/2),}
and the general polygamma function as
ψ
p
(
n
)
(
a
)
=
∂
n
log
Γ
p
(
a
)
∂
a
n
=
∑
i
=
1
p
ψ
(
n
)
(
a
+
(
1
−
i
)
/
2
)
.
{\displaystyle \psi _{p}^{(n)}(a)={\frac {\partial ^{n}\log \Gamma _{p}(a)}{\partial a^{n}}}=\sum _{i=1}^{p}\psi ^{(n)}(a+(1-i)/2).}
Calculation steps
Γ
p
(
a
)
=
π
p
(
p
−
1
)
/
4
∏
j
=
1
p
Γ
(
a
+
1
−
j
2
)
,
{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+{\frac {1-j}{2}}),}
it follows that
∂
Γ
p
(
a
)
∂
a
=
π
p
(
p
−
1
)
/
4
∑
i
=
1
p
∂
Γ
(
a
+
1
−
i
2
)
∂
a
∏
j
=
1
,
j
≠
i
p
Γ
(
a
+
1
−
j
2
)
.
{\displaystyle {\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\sum _{i=1}^{p}{\frac {\partial \Gamma (a+{\frac {1-i}{2}})}{\partial a}}\prod _{j=1,j\neq i}^{p}\Gamma (a+{\frac {1-j}{2}}).}
∂
Γ
(
a
+
(
1
−
i
)
/
2
)
∂
a
=
ψ
(
a
+
(
i
−
1
)
/
2
)
Γ
(
a
+
(
i
−
1
)
/
2
)
{\displaystyle {\frac {\partial \Gamma (a+(1-i)/2)}{\partial a}}=\psi (a+(i-1)/2)\Gamma (a+(i-1)/2)}
it follows that
∂
Γ
p
(
a
)
∂
a
=
π
p
(
p
−
1
)
/
4
∏
j
=
1
p
Γ
(
a
+
(
1
−
j
)
/
2
)
∑
i
=
1
p
ψ
(
a
+
(
1
−
i
)
/
2
)
=
Γ
p
(
a
)
∑
i
=
1
p
ψ
(
a
+
(
1
−
i
)
/
2
)
.
{\displaystyle {\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2)\sum _{i=1}^{p}\psi (a+(1-i)/2)=\Gamma _{p}(a)\sum _{i=1}^{p}\psi (a+(1-i)/2).}
References
![{\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left[a+(1-j)/2\right].}](/media/api/rest_v1/media/math/render/svg/4dc9bcc58164ece0de7aef7b0ba2b1a0925a1a04)
![{\displaystyle \Gamma _{p}(a)=\pi ^{(p-1)/2}\Gamma (a)\Gamma _{p-1}(a-{\tfrac {1}{2}})=\pi ^{(p-1)/2}\Gamma _{p-1}(a)\Gamma [a+(1-p)/2].}](/media/api/rest_v1/media/math/render/svg/842b762835010dccfea7a4e848cefcecb30f3e8e)
