Divided differences and complex variations of multiple zeta-star values

Abstract

The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper, we study the corresponding limiting set for finite multiple star harmonic sums. Using the theory of divided differences, we construct a natural complex analytic interpolation of finite multiple star harmonic sums. For real $s>1$, we analyze the range of this interpolation in detail and prove a finite zeta-star correspondence. In the complex case, we formulate an injectivity conjecture, which may be viewed as the complex variation of zeta-star correspondence for multiple zeta-star values.