Positivity properties of some special matrices

It is shown that for positive real numbers $0<λ_{1}<\dots<λ_{n}$, $\left[\frac{1}{β({λ_i}, {λ_j})}\right]$, where $β(\cdot,\cdot)$ denotes the beta function, is infinitely divisible and totally positive. For $\left[\frac{1}{β({i},{j})}\right]$, 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.

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