Computable structural formulas for the distribution of the \(\beta\)-Jacobi edge eigenvalues (Q6158137)

The so-called \(\beta\)-Jacobi ensemble can been seen as a generalization of the beta distribution in several dimensions. In fact, it represents a joint density of the set of eigenvalues for certain random matrices models. An important question is to know statistical information of the largest and the smallest eigenvalues, as well as their density probability functions. Such density functions can be obtained using the \(\beta\)-Jacobi ensemble after integrating over a certain domain. In this paper the authors propose some algorithms to compute such an integral under some specific selection of parameters, which helps to carry on the analysis. The proposed algorithms are based on some differential-difference systems of equations that were known before for some related integrals of the \(\beta\)-Jacobi ensemble.