Multivariate gamma function
Encyclopedia
In mathematics
, the multivariate Gamma function, Γp(·), is a generalization of the 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
where S>0 means S is positive-definite. The other one, more useful in practice, is
From this, we have the recursive relationships:
Thus
and so on.
and the general polygamma function as 
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the multivariate Gamma function, Γp(·), is a generalization of the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. It is useful in multivariate statistics
Multivariate statistics
Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...
, appearing in the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of the Wishart and Inverse Wishart distributions.
It has two equivalent definitions. One is
where S>0 means S is positive-definite. The other one, more useful in practice, is
From this, we have the recursive relationships:
Thus
and so on.
Derivatives
We may define the multivariate digamma function as and the general polygamma function as Calculation steps
- Since
, it follows that 
. - Because
(by definition of the digamma function 
), we have 