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{{Short description|Multivariate generalization of the gamma function}}
In [[mathematics]], the '''multivariate gamma function''' Γ<sub>''p''</sub> is a generalization of the [[gamma function]]. It is useful in [[multivariate statistics]], appearing in the [[probability density function]] of the [[Wishart distribution|Wishart]] and [[inverse Wishart distribution]]s, and the [[matrix variate beta distribution]].<ref>{{Cite journal|last=James|first=Alan T.|date=June 1964|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples|url=https://projecteuclid.org/euclid.aoms/1177703550|journal=The Annals of Mathematical Statistics|language=en|volume=35|issue=2|pages=475–501|doi=10.1214/aoms/1177703550|issn=0003-4851|doi-access=free}}</ref>

It has two equivalent definitions. One is given as the following integral over the <math>p \times p</math> [[positive-definite matrix|positive-definite]] real matrices:

:<math>
\Gamma_p(a)=
\int_{S>0} \exp\left(
-{\rm tr}(S)\right)\,
\left|S\right|^{a-\frac{p+1}{2}}
dS,
</math>
where <math>|S|</math> denotes the determinant of <math>S</math>. The other one, more useful to obtain a numerical result is:

:<math>
\Gamma_p(a)=
\pi^{p(p-1)/4}\prod_{j=1}^p
\Gamma(a+(1-j)/2).
</math>
In both definitions, <math>a</math> is a complex number whose real part satisfies <math>\Re(a) > (p-1)/2</math>. Note that <math>\Gamma_1(a)</math> reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for <math>p\ge 2</math>:
:<math>
\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).
</math>

Thus

* <math>\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)</math>
* <math>\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)</math>

and so on.

This can also be extended to non-integer values of <math>p</math> with the expression:

<math>\Gamma_p(a)=\pi^{p(p-1)/4} \frac{G(a+\frac{1}2)G(a+1)}{G(a+\frac{1-p}2)G(a+1-\frac{p}2)}</math>

Where G is the [[Barnes G-function]], the [[indefinite product]] of the [[Gamma function]].

The function is derived by Anderson<ref>{{Cite book|last=Anderson|first=T W|title=An Introduction to Multivariate Statistical Analysis|publisher=John Wiley and Sons|year=1984|isbn=0-471-88987-3|location=New York|pages=Ch. 7}}</ref> from first principles who also cites earlier work by [[John Wishart (statistician)|Wishart]], [[Prasanta Chandra Mahalanobis|Mahalanobis]] and others.

There also exists a version of the multivariate gamma function which instead of a single complex number takes a <math>p</math>-dimensional vector of complex numbers as its argument. It generalizes the above defined multivariate gamma function insofar as the latter is obtained by a particular choice of multivariate argument of the former.<ref>{{cite web|url=https://dlmf.nist.gov/35|title=Chapter 35 Functions of Matrix Argument|work=[[Digital Library of Mathematical Functions]]|author=[[Donald Richards (statistician)|D. St. P. Richards]]|date=n.d.|access-date=23 May 2022}}</ref>

== Derivatives ==

We may define the multivariate [[digamma function]] as
:<math>\psi_p(a) = \frac{\partial \log\Gamma_p(a)}{\partial a} = \sum_{i=1}^p \psi(a+(1-i)/2) ,</math>
and the general [[polygamma function]] as
:<math>\psi_p^{(n)}(a) = \frac{\partial^n \log\Gamma_p(a)}{\partial a^n} = \sum_{i=1}^p \psi^{(n)}(a+(1-i)/2).</math>

=== Calculation steps ===

* Since
::<math>\Gamma_p(a) = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma\left(a+\frac{1-j}{2}\right),</math>
:it follows that
::<math>\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\sum_{i=1}^p \frac{\partial\Gamma\left(a+\frac{1-i}{2}\right)}{\partial a}\prod_{j=1, j\neq i}^p\Gamma\left(a+\frac{1-j}{2}\right).</math>

* By definition of the [[digamma function]], ψ,
::<math>\frac{\partial\Gamma(a+(1-i)/2)}{\partial a} = \psi(a+(i-1)/2)\Gamma(a+(i-1)/2)</math>

:it follows that
::<math>
\begin{align}
\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) \\[4pt]
& = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).
\end{align}
</math>

{{more footnotes|date=May 2012}}

==References==
{{Reflist}}
* 1. {{cite journal
|title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
|last=James |first=A.
|journal=[[Annals of Mathematical Statistics]]
|volume=35 |issue=2 |year=1964 |pages=475–501
|doi=10.1214/aoms/1177703550 |mr=181057 | zbl = 0121.36605
|doi-access=free }}
* 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

[[Category:Gamma and related functions]]

Multivariate gamma function - Revision history

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