## Taiwanese Journal of Mathematics

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

Chung-Yie Chang

#### 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 $\delta_{mm=1}$.

#### Article information

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

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874418

Digital Object Identifier
doi:10.11650/tjm.20.2016.4325

Mathematical Reviews number (MathSciNet)
MR3462864

Zentralblatt MATH identifier
1366.11097

#### Citation

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

#### References

• \labelbib01 B. C. Berndt, Ramanujan's Notebooks, Part I and II, Springer-Verlag, New York 1985, 1989. http://dx.doi.org/10.1007/978-1-4612-4530-8
• \labelbib02 D. Borwein, J. M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277–294.
• \labelbib03 R. E. Crandall and J. P. Buhler, On the evaluation of Euler sums, Experiment. Math. 3 (1994), no. 4, 275–285.
• \labelbib04 M. Eie, Topics in Number Theory, Monographs in Number Theory 2, World Scientific, Hackensack, NJ, 2009.
• \labelbib05 ––––, The Theory of Multiple Zeta Values with Applications in Combinatorics, Monographs in Number Theory 7, World Scientific, Hackensack, NJ, 2013.
• \labelbib06 M. Eie, W.-C. Liaw and Y. L. Ong, A restricted sum formula among multiple zeta values, J. Number Theory 129 (2009), no. 4, 908–921.
• \labelbib08 M. Eie and C.-S. Wei, Generalizations of Euler decomposition and their applications, J. Number Theory 133 (2013), no. 8, 2475–2495.
• \labelbib09 M. Eie, T.-Y. Lee and Y. L. Ong, Applications of shuffle products of multiple zeta values in combinatorics, J. Comb. Number Theory 4 (2013), no. 3, 145–160.
• \labelbib11 A. Granville, A decomposition of Riemann's zeta-function, in: Analytic Number Theory, (Kyoto, 1996), 95–101; London Math. Soc. Lecture Note Ser. 247, Cambridge Univ. Press, Cambridge, 1997.
• \labelbib10 L. Guo and B. Xie, Weighted sum formula for multiple zeta values, J. Number Theory 129 (2009), no. 11, 2747–2765.
• \labelbib07 M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275–290.
• \labelbib13 Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory 74 (1999), no. 1, 39–43.
• \labelbib12 Y. L. Ong, M. Eie and W.-C. Liaw, On generalizations of weighted sum formulas of multiple zeta values, Int. J. Number Theory 9 (2013), no. 5, 1185-1198.