Multivariate gamma function
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, and the matrix variate beta distribution.[1]
It has two equivalent definitions. One is given as the following integral over the positive-definite real matrices:
(note that reduces to the ordinary gamma function). The other one, more useful to obtain a numerical result is:
From this, we have the recursive relationships:
Thus
and so on.
This can also be extended to non-integer values of p with the expression:
Where G is the Barnes G-function, the indefinite product of the Gamma function.
Derivatives
We may define the multivariate digamma function as
and the general polygamma function as
References
- James, Alan T. (June 1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". The Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. ISSN 0003-4851.
- 1. James, A. (1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. MR 0181057. Zbl 0121.36605.
- 2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.