Monotonicity of Coefficients of Interpolated Multiple Zeta Values

Abstract

Let $ζ^t(\boldsymbol{k})=\sum_{d=1}^{r}S_d(\boldsymbol{k})t^{r-d}$ be the interpolated multiple zeta value of an admissible index $\boldsymbol{k}$ of depth $r$. We show the monotonicity property of coefficients $S_1(\boldsymbol{k})\ge\cdots\ge S_r(\boldsymbol{k})>0$. 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 $ζ^{-1}(\boldsymbol{k})$ and locate the zeros of $ζ^t(\boldsymbol{k})$ in the closed unit disk.