Monotonicity of Coefficients of Interpolated Multiple Zeta Values

Abstract

Let ${\zeta }^t(\mathbf{k})={\sum }_{d=1}^r{S}_d(\mathbf{k})t^{r-d}$ be the interpolated multiple zeta value of an admissible index $\mathbf{k}$ of depth $r$. We show the monotonicity property of coefficients $S_1(\mathbf{k})\ge \cdots \ge S_r(\mathbf{k})>0$. In the proof, we represent the coefficients as integrals over a common measure space and compares the integrands pointwise. As consequences, we determine the sign and vanishing of ${\zeta }^{-1}(\mathbf{k})$ and locate the zeros of ${\zeta }^t(\mathbf{k})$ in the closed unit disk.

arXiv comment8 pages