Taiwanese Journal of Mathematics

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

Chung-Yie Chang

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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 $\delta_{mm=1}$.

Article information

Taiwanese J. Math., Volume 20, Number 1 (2016), 13-24.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 40A25: Approximation to limiting values (summation of series, etc.) {For the Euler-Maclaurin summation formula, see 65B15} 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M01 33E20: Other functions defined by series and integrals

multiple zeta values shuffle product Euler decomposition theorem


Chang, Chung-Yie. Shuffle Product Formulas of Two Multiples of Height-one Multiple Zeta Values. Taiwanese J. Math. 20 (2016), no. 1, 13--24. doi:10.11650/tjm.20.2016.4325. https://projecteuclid.org/euclid.twjm/1498874418

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