Appendix A — Special functions

A.1 Gamma function

The gamma function is defined for \(x > 0\) by the integral \[ \Gamma(x) = \int_{s>0} e^{-s} s^{x-1} ds \] where \(s\) is a positive scalar.

The gamma function \(\Gamma(x+1)\) provides a continuous extension of the factorial \(x!\), with \(\Gamma(x+1) = x!\) for any nonnegative integer \(x\).

The multivariate gamma function is defined for \(x > (d-1)/2\) by the integral \[ \Gamma_d(x) = \int_{S>0} \exp(-\trace(S)) \det(S)^{x-(d+1)/2} dS \] where \(\bS\) is a real symmmetric positive definite matrix.

The integral can be computed in terms of gamma functions as \[ \Gamma_d(x) = \pi^{d (d-1)/4} \prod_{j=1}^d \Gamma(x - (j-1)/2) \]

For \(d=1\) the multivariate gamma function reduces to the standard gamma function, \(\Gamma_1(x) = \Gamma(x)\).

A.2 Digamma function

The digamma function is the first derivate of the logarithm of the gamma function: \[ \psi^{(0)}(x) =\frac{d}{dx} \log \Gamma(x) \]

The multivariate digamma function is the first derivative of the logarithm of the multivariate gamma function: \[ \psi^{(0)}_d = \frac{d}{dx} \log \Gamma_d(x) = \sum_{i=1}^d \psi^{(0)}(x - (i-1)/2) \]

For \(d=1\) the multivariate digamma function reduces to the standard digamma function, \(\psi^{(0)}_1(x) = \psi^{(0)}(x)\).

A.3 Beta function

The beta function is given by \[ B(\alpha_1, \alpha_1) = \frac{ \Gamma(\alpha_1)\, \Gamma(\alpha_2)}{\Gamma(\alpha_1+\alpha_2)} \]

The multivariate beta function is given by \[ B(\balpha) = \frac{ \prod_{k=1}^K \Gamma(\alpha_k) }{\Gamma( \sum_{k=1}^K \alpha_k )} \]

For two groups (\(K=2\)) the multivariate beta function reduces to the conventional beta function, \(B((\alpha_1, \alpha_2)^T) = B(\alpha_1, \alpha_2)\).