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Positivity properties of some special matrices. (English) Zbl 1435.15024

Summary: It is shown that for positive real numbers \(0 < \lambda_1 < \dots < \lambda_n\), \([\frac{1}{\beta(\lambda_i,\lambda_j)}]\), where \(\beta(\cdot, \cdot)\) denotes the beta function, is infinitely divisible and totally positive. For \([\frac{1}{\beta(i,j)}]\), the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let \(\mathfrak{w}(n)\) be the \(n\) th Bell number. It is proved that \([\mathfrak{w}(i + j)]\) 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.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A23 Factorization of matrices
42A82 Positive definite functions in one variable harmonic analysis
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