臺灣博碩士論文加值系統

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 dierent integral ........24
5. Evaluation of the combined generating function ..................29

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