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 For an $r$-triple of positive integers $\alpha=(\alpha_1, \alpha_2, \cdots, \alpha_r)$with $\alpha\geq2$ a multiple zeta value of depth $r$ and weight $\vert\alpha\vert=\alpha_1 + \alpha_2 + \ldots + \alpha_r$ is defined as\begin{align*}\zeta(\alpha_1, \alpha_2, \cdot, \alpha_r)=\sum_{1\leq n_1 \leq n_2\leq \cdots \leq n_r}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r}.\end{align*}The shuffle product of two multiple zeta values of weight $m$ and $n$ will produce $\binom{m+n}{m}$ multiple zeta values of weight $m+n$. Based on such a fact, we are able to evaluate the generating function\begin{align*}\sum_{j=0}^p\sum_{\ell=0}^q \binom{p+q+4}{j+q-\ell+2}\mu^j\lambda^{\ell}\end{align*}and\begin{align*}\sum_{j=0}^k\sum_{\ell=0}^r \binom{k+r+4}{j+r-\ell+2} \binom{j+r-\ell+1}{r-\ell}\binom{k-j+\ell+1}{k-j}\mu^j\lambda^{\ell+1}\end{align*}in terms of certain simple polynomials. After separating the coefficients of $\mu^j\lambda^{\ell}$ we get some interesting combinatorial identities, such as, for integers $p$, $q$, $j$, $\ell$ with $0\leq j \leq p$ and $0\leq \ell \leq q$,\begin{align*}\binom{p+q+4}{j+\ell+2}=&\sum_{a+b=p}\binom{a}{j}\binom{b+q+3}{q-\ell+1} + \sum_{a+b=p}\binom{a}{p-j}\binom{b+q+3}{\ell+1}.\end{align*}
 ContentsAbstract 11. Introduction................................................... ..32. Drinfeld integrals and shue relations of multiple zeta values.... 73. Evaluation of the function I1 and I2 ............................154. The same generating function through a di erent integral ........245. Evaluation of the combined generating function ..................29
 [1] 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.[2] D. Bowman and D. M. Bradley, Multiple polylogarithms: a brief survey, Con-temp. Math. 291 (2001), 71-92.[3] Y. Ohno, A generalization of the duality and sum formulae on the multiple zeta values, J. Number Thoery 74 (1999), no. 1, 39-43.[4] J. M. Borwein and R. Girgensohn, Evaluation of triple Euler sums, Electorn, J. Combin. 3 (1996), no. 1, 1-27.[5] A. Granville, A decomposition of Riemann's zeta-function, Analytic number theory (Kyoto,1996), 95-101.[6] C. Markett, Triple sums and the Riemann zeta function, J. Number Theory 48(1994), no. 2, 113-132.[7] M. E. Ho man, Multiple harmonic series, Paci c J. Math. 152 (1992), no. 2, 275-290.[8] B. C. Berndt, Ramanujan's Notebooks, Part I and II, Springer-Verlag, New York, 1985, 1989.[9] L. Euler, Opera Ommia, Ser 1, Vol. XV (Teubner, Berlin 1917), 217-267.3
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 1 Algebraic Relations for Multiple Zeta Values Through Shuffle Product Formulas 2 A New Combinatorial Identity from Multiple Zeta Values 3 Multiple Zeta Values And Their Applications In Combinatorics 4 一些zeta函數結合multiple zeta values 5 論多重zeta值的廣義組合等式 6 Applications of the hypergeometric distribution to shuffle relations of multiple zeta values 7 透過Drinfeld積分計算多重zeta值高度為1 8 求和公式和多重zeta值的ShuffleRelations 9 多重zeta值上的代數關係式 10 Some Combinatorial Identities Produced from Shuffle Products of Two Sums of Multiple Zeta Values 11 Some relations between multiple zeta values and multiple zeta-star values 12 運用三角Bézier曲片逼近三角網格 13 修正型自組特徵映射神經網路應用於離散數據的聚類 14 A survey of a recent of Chen-Li-Ou on theclassification of solution of an integral equation 15 濺鍍薄膜之光學常數與界面應力

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