Positivity properties of some special matrices

Abstract

It is shown that for positive real numbers 0<λ1<…<λn, [1β(λi,λj)], where β(⋅,⋅) denotes the beta function, is infinitely divisible and totally positive. For [1β(i,j)], the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i+j)] 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.