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研究生:李坤憲
研究生(外文):Li Kunhsien
論文名稱:有關多重zeta值的組合結果
論文名稱(外文):Combinatorial Consequences Relating Multiple Zeta Values
指導教授:余文卿余文卿引用關係
指導教授(外文):Eie Minking
口試委員:翁耀臨廖文欽江謝宏任
口試委員(外文):Weng Yao-LinLiaw WenchinChiang-Hsieh Hungjen
口試日期:2012-06-15
學位類別:碩士
校院名稱:國立中正大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
語文別:英文
論文頁數:38
中文關鍵詞:zeta 值組合式洗牌過程Drifeld 積分
外文關鍵詞:zeta valuescombinedshuffle relationDrifeld integral
相關次數:
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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*}
Contents
Abstract 1
1. Introduction................................................... ..3
2. Drinfeld integrals and shue relations of multiple zeta values.... 7
3. Evaluation of the function I1 and I2 ............................15
4. The same generating function through a di erent integral ........24
5. 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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