A matrix-variate extension of inverted Dirichlet integral (Q2484106)

The inverted Dirichlet or type-2 Dirichlet density, defined by \[ f_1(x_1,\dots,x_k)=\frac{\Gamma(\alpha_1+\dots+\alpha_{k+1})} {\Gamma(\alpha_1)\dots\Gamma(\alpha_{k+1})} x_1^{\alpha_1-1}\dots x_k^{\alpha_k-1} (1+x_1+\dots+x_k)^{-(\alpha_1+\dots+\alpha_k)}, \] where \(0\leq x_j<\infty\), \(j=1,\dots,k\) and \(f_1(x_1,\dots,x_k)=0\), elsewhere, is extended by the authors in terms of matrix-variate type-2 density given by \[ \begin{multlined} f_2(X_1,\dots,X_k)= \frac{\Gamma_p(\alpha_1+\dots+\alpha_{k+1})} {\Gamma_p(\alpha_1)\dots\Gamma_p(\alpha_{k+1})}\\ |X_1|^{\alpha_k-\frac{p+1}{2}}\dots |X_k|^{\alpha_k-\frac{p+1}{2}}\;|I+X_1+\dots+X_k|^{-(\alpha_1+\dots+\alpha_{k+1})},\end{multlined}\tag{\(*\)} \] where \(X_j=X_j'>0\), \(j=1,\dots,k\), \(\text{Re}(\alpha_j)>\frac{p-1}{2}\), \(j=1,\dots,k+1\), and zero elsewhere. Let \(X_j=X_j'>0\) be \(p\times p\) real symmetric definite matrices, then their joint density provides the generalization of the density model (\(*\)) given by the authors in terms of the generalized real-type-2 Dirichlet density in the form \[ \begin{multlined} f_3(X_1,\dots,X_k)=\left(\prod_{j=1}^k \frac{\Gamma_p(\alpha_j)\Gamma_p (\alpha{j+1}+\dots+\alpha_{k+1} +\beta_j+\dots+\beta_k)}{\Gamma_p(\alpha{j}+\dots+\alpha_{k+1} +\beta_j+\dots+\beta_k)}\right) \\ \times |X_1|^{\alpha_1-\frac{p+1}{2}}\dots|X_k|^{\alpha_k-\frac{p+1}{2}}\;|I+X_2+\dots+X_k|^{\beta_1}\;|I+X_3+\dots+X_k|^{\beta_2} \\ \times \dots\;|I+X_1+\dots+X_k|^{-(\alpha_1+\dots+\alpha_{k+1}+\beta_1+\dots+\beta_k)} \end{multlined} \] under certain convergence conditions, and \(f(X_1,\dots,X_k)=0\) elsewhere. Its various properties are also established.