# 臺灣博碩士論文加值系統

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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 asimplex of weight dimension and the shuffle product of two multiple zeta valuescan be defined. In this dissertation I shall provide a number of algebraicrelations among multiple zeta values using a modified shuffle product formulato certain integrals. Furthermore, the shuffle product of two multiple zetavalues 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 zetavalues in relations produced from the shuffle product of two particularmultiple 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 vAbstract viiIntroduction ix 0.1 Historical Background and Notations . . . ix 0.2 Overview . . . xii1 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 . . . 112 Drinfel'd Integral Representations 17 2.1 Drinfel'd Integral Representations and Shue 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 . . . 223 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 . . . 754 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 Shue Relation Concerning Multiple Zeta Values of Height Three . . . 93 4.4 A Generalization of Ohno{Zudilin's Relation . . . 965 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 . . . 119Bibliography 123