# What is the fisher information matrix of the multivariate t distribution?

$$\newcommand{\bx}{\mathbf{x}}$$ $$\newcommand{\bSigma}{\boldsymbol{\Sigma}}$$ $$\newcommand{\bE}{\mathbf{E}}$$ $$\newcommand{\bD}{\mathbf{D}}$$

Consider the multivariate central t distribution with p.d.f. \begin{align} f(\bx| \nu, \bSigma) = \frac{\nu^{\frac{\nu}{2}}\Gamma(\frac{\nu+p}{2})}{\pi^\frac{p}{2}\Gamma(\frac{\nu}{2})} |\bSigma|^{-\frac{1}{2}} \left(\nu + \bx'\bSigma^{-1}\bx\right)^{-\frac{\nu + p}{2}}, \end{align} where $$\nu$$ is a scalar parameter and $$\bSigma$$ is a $$p$$ by $$p$$ symmetric positive definite parameter matrix.

The log-likelihood function is $$\mathcal{L(\bSigma, \nu| \bx)} = c(\nu) -\frac{1}{2} log(\left|\bSigma\right|) - \frac{\nu + p}{2} log(\nu + \bx'\bSigma^{-1}\bx).$$

Does anyone know the Fisher information matrix w.r.t. $$\bSigma$$, that is $$- \bE \left[ \frac{\partial^2 \mathcal{L}}{\partial vech(\bSigma) \partial vech(\bSigma)'}\right]$$ or $$\bE \left[ \frac{\partial \mathcal{L}}{\partial vech(\bSigma)} \left(\frac{\partial \mathcal{L}}{\partial vech(\bSigma)}\right)' \right]$$, or where I could find it?