Taylor expansions of analytic functions related to \((1+z)^ x-1\)
The author studies the MacLaurin coefficients of functions related to \((1+z)^ x-1\), where \(x\in {\mathbb{C}}\) is arbitrary. He gives recurrence relations and explicit representation formulas, some of which depend on the Stirling numbers \(S(n,k)\) and \(s(n,k)\) of the second and the first kind respectively, which are defined by \[ (k!)^{-1}(e^ z-1)^ k=\sum^{\infty}_{n=k}S(n,k)z^ n/n! \] and \[ (k!)^{-1}(\ell n(1+z))^ k=\sum^{\infty}_{n=k}s(n,k)z^ n/n!. \]
scientific article; zbMATH DE number 4053978