| Summary: | 博士 === 國立中正大學 === 數學研究所 === 100 === 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
$j \leq n$.
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