A random matrix decimation procedure relating $\beta =2\mathrm{/}(r+1)$ to $\beta =2(r+1)$

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta =2\mathrm{/}(r+1)$ is equal to that of another $\beta$-ensemble with $\beta =2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.