The derived set of multiple star polylogarithms

Source abstract

The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence and hence admits a normalized Hausdorff--Stieltjes representation. Both the shifted and the unshifted functions, as well as their infinite-depth limits, are shown to be univalent on the half-plane $\mathrm{Re}\,z \frac12$. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane $\mathrm{Re}\,w\geq\frac12$.