Positivity properties of some special matrices

It is shown that for positive real numbers 0 < lambda(1) < ... < lambda(n), [1/beta(lambda i,lambda j)], where beta(.,.) denotes the beta function, is infinitely divisible and totally positive. For [1/beta(i,j)], the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let to (n) be the nth Bell number. It is proved that [to(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. (C) 2020 Elsevier Inc. All rights reserved.