Existence conditions of permanental and multivariate negative binomial distributions (Q682281)

We say that a matrix \(A\) is \(\beta\)-positive definite if the multivariate Taylor expansion in \(z_1^{n_1}\dots z_d^{n_d}\) of \(\det(I-Z A)^{-\beta}\), where \(Z\) is a diagonal matrix with elements \(z_1\), \(\dots\), \(z_d\), has only nonnegative coefficients. In this manuscript, the authors correct some older results about the \(\beta\)-positive definite matrix and \(\beta\)-permanental matrix and introduce some new results. They first correct the criteria introduced in [\textit{R. C. Griffiths} and \textit{R. K. Milne}, J. Multivariate Anal. 22, 13--23 (1987; Zbl 0618.62060)] and use it to obtain some results about \(\beta\)-positivity. The rest of the manuscript is dedicated on \(\beta\)-permanental matrix of dimensions below and above \(3\).