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研究生:李東洋
研究生(外文):Lee, Tung-Yang
論文名稱(外文):Algebraic Relations for Multiple Zeta Values Through Shuffle Product Formulas
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:余文卿廖文欽江謝宏任翁耀臨楊富堯
口試委員(外文):Eie, MinkingLiaw, Wen-ChinChiang-Hsieh, Hung-JenOng, Yao LinYang, Fu-Yao
口試日期:2012-06-15
學位類別:博士
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:141
中文關鍵詞:多重 Zeta 值
外文關鍵詞:Multiple Zeta Values
相關次數:
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For a multi-index $\mfa = (\seq{a}{1}{2}{p})$ of positive integers with $a_{p}
\geq 2$, a multiple zeta value of depth $p$ and weight $\av{\mfa} =
\fsum{a}{1}{2}{p}$ or $p$-fold Euler sum is defined to be
\[
\zeta(\seq{a}{1}{2}{p})
= \sum_{1 \leq n_{1} < n_{2} < \cdots < n_{p}}
n_{1}^{-a_{1}} n_{2}^{-a_{2}} \cdots n_{p}^{-a_{p}},
\]
which is a natural generalization of the classical Euler sum
\[
S_{a, b}
= \sum_{k=1}^{\infty} \frac{1}{k^{b}} \sum_{j=1}^{k} \frac{1}{j^{a}}, \quad
a, b \in \bn, \quad b \geq 2.
\]

Multiple zeta values can be expressed as Drinfel'd iterated integrals over a
simplex of weight dimension and the shuffle product of two multiple zeta values
can be defined. In this dissertation I shall provide a number of algebraic
relations among multiple zeta values using a modified shuffle product formula
to certain integrals. Furthermore, the shuffle product of two multiple zeta
values of weight $m$ and $n$, respectively, will produce $\binom{m+n}{m}$
multiple zeta values of weight $m+n$. By counting the number of multiple zeta
values in relations produced from the shuffle product of two particular
multiple zeta values, we obtain many specific combinatorial identities such as
\[
\binom{m+n+4}{i+j+2}
= \sum_{m_{1}+m_{2}=m} \brac{\binom{m_{1}}{i} \binom{m_{2}+n+3}{j+1}
+ \binom{m_{1}}{m-i} \binom{m_{2}+n+3}{n-j+1}}
\]
and
\begin{align*}
\binom{i+j}{i} \binom{m+n+4}{i+j+2}
&= \binom{i+j}{i} \brac{\binom{m+j+3}{i+j+2} + \binom{i+n+2}{i+j+2}} \\
&\quad + \sum_{m_{1}+m_{2}=m} \sum_{n_{1}+n_{2}=n}
\binom{n_{1}+i-m_{2}}{n_{1}} \binom{n_{1}+m_{1}+2}{m-i+1}
\binom{m_{2}+j-n_{1}}{m_{2}} \binom{m_{2}+n_{2}+1}{n-j},
\end{align*}
where $(m, n, i, j)$ is a quadruple of nonnegative integers with $i \leq m$ and
$j \leq n$.
Acknowledgement v

Abstract vii

Introduction ix
0.1 Historical Background and Notations . . . ix
0.2 Overview . . . xii

1 Classical Euler Sums, Triple Zeta Values, and Their Analogues 1
1.1 Evaluations of Classical Euler Sums . . . 1
1.2 Evaluations of Extended Euler Sums . . . 3
1.3 Parametric Euler Sums . . . 5
1.4 Evaluations of Euler Sums with a Dirichlet Character . . . 7
1.5 Evaluations of Multiple Zeta Values of Depth Three and Height One . . . 9
1.6 Evaluations of Triple Zeta Values . . . 11

2 Drinfel'd Integral Representations 17
2.1 Drinfel'd Integral Representations and Shue Products of Multiple Zeta Values . . . 17
2.2 A Generating Function of Multiple Zeta Values of Height One . . . 21
2.3 Proofs of the Sum Formula and the Restricted Sum Formula . . . 22

3 Weighted and Double Weighted Sum Formulas Concerning Multiple Zeta Values 25
3.1 Euler's Decomposition Formula and Its Generalization . . . 25
3.2 Weighted Sum Formulas and Ohno{Zudilin's Relation . . . 27
3.3 Two Examples . . . 30
3.4 A Sort of Speci c Weighted Sum Formula: Type I . . . 32
3.5 Two Simple Examples . . . 35
3.6 A Sort of Speci c Double Weighted Sum Formula: Type II . . . 38
3.7 Double Weighted Sum Formula: Type III . . . 43
3.8 Double Weighted Sum Formula: Type IV . . . 48
3.9 Double Weighted Sum Formula: Type V . . . 52
3.10 Two Speci c Double Weighted Sum Formulas . . . 57
3.11 A Weighted Sum Formula of Double Zeta Values . . . 63
3.12 Relations Between Multiple Zeta Values and Theorem 3.2 . . . 68
3.13 Applications of Vandermonde's Convolution to Multiple Zeta Values . . . 70
3.14 Double Weighted Sum Formula: Type VI . . . 75

4 Identities Involving Multiple Zeta Values of Height Two 79
4.1 Three Important Examples . . . 79
4.2 Double Weighted Sum Formulas Concerning Multiple Zeta Values of Height Two . . . 89
4.3 A Shue Relation Concerning Multiple Zeta Values of Height Three . . . 93
4.4 A Generalization of Ohno{Zudilin's Relation . . . 96

5 Combinatorial Identities Relating Multiple Zeta Values 99
5.1 Examples . . . 99
5.2 More Examples Coming from Section 3.13 . . . 103
5.3 A Generalization of Pascal's Identity . . . 109
5.4 A Generating Function for $\binom{i+j}{i} \binom{m+n+4}{i+j+2}$ and Its Decomposition . . . 112
5.5 Some Particular Combinatorial Identities . . . 117
5.6 A Generating Function for $(i+1) (m-i+1) \binom{m+n+4}{i+j+2}$ and Its Decomposition . . . 119

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