Positivity properties of some special matrices

Abstract

It is shown that for positive real numbers $0<{\lambda }_1<\cdots <{\lambda }_n$, $\left[\frac{1}{\beta ({\lambda }_i,{\lambda }_j)}\right]$, where $\beta (\cdot ,\cdot )$ denotes the beta function, is infinitely divisible and totally positive. For $\left[\frac{1}{\beta (i,j)}\right]$, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let $\mathfrak{w}(n)$ be the $n$th Bell number. It is proved that $\left[\mathfrak{w}(i+j)\right]$ is a totally positive matrix but is infinitely divisible only upto order $4$. It is also shown that the symmetrized Stirling matrices are totally positive.

9 pages