Shuffle Product Formulas of Two Multiples of Height-one Multiple Zeta Values

Abstract

The classical Euler decomposition theorem expressed a product of two Riemann zeta values $\zeta(p) \zeta(q)$ as a sum of $\binom{p+q}{p}$ Euler double sums of weight $p+q$.

As a generalization of Euler decomposition theorem, we shall perform the shuffle product of two multiples of height-one multiple zeta values\[ \binom{j+m}{m} \zeta(\{1\}^{j+m-1}, r-\ell+2) \quad \textrm{and} \quad \binom{k-j+n}{n} \zeta(\{1\}^{k-j+n-1}, \ell+2)\]with positive integers $m, n$ and integers $k, j, r, \ell$ such that $0 \leq j \leq k$, $0 \leq \ell \leq r$. Then we applied the resulted shuffle relation to produce weighted sum formulas such as\begin{align*} & \quad (k+1) \sum_{|\boldsymbol{\alpha}|=k+r+2} \zeta(\alpha_0+1, \ldots, \alpha_{k+1}+1) 2^{\alpha_{k+1}} \\ & \quad + 2\sum_{|\boldsymbol{\alpha}|=k+r+1} \zeta(1, \alpha_0+1, \ldots, \alpha_k+1) 2^{\alpha_k - \delta_{0k}} \\ & = \frac{1}{2} \sum_{j=0}^k \sum_{\substack{\ell=0 \\ \ell: \textrm{even}}}^r (-1)^j (j+1) (k-j+1) \zeta(\{1\}^j, r-\ell+2) \zeta(\{1\}^{k-j}, \ell+2)\end{align*}when both $k$ and $r$ are even. Here $\delta_{mn} = 0$ unless $m=n$ and $\delta_{mm=1}$.