05 Stirling-matrices St1 and St2

Abstract: The Stirling-matrices occur as other initimate and basic relatives of the ZV-(Vandermonde) matrix. Variants transform powerseries to exponentialseries and conversely. Using finite sizes they or their scaled variants give rational approximations to logarithms and exponentials. The most striking property for me is, that they are eigenmatrices of the Bernoullian-matrix Gp, which sums geometric series to zeta-type series of any like powers to any finite number of terms. Most of the formulae here are heuristic findings (although in the meantime I found most of the formulas in textbooks and online-references). The focus in my recent study was primarily at the binomial- and the Gp-matrix; but I expect to understand more details of these matrices when analyzing the Stirling-matrices intensely. Contents