| Summary: | 碩士 === 國立中正大學 === 數學系應用數學研究所 === 107 === Abstract
In this thesis, we introduce the method of using an integral to represent a sum of restricted sums. Also we, via the generating dual function, obtain a weighted sum formulas of different depths of multiple zeta values. If
\begin{eqnarray*}
T(m,n)&:=&\frac{1}{m!n!}\int_{E_2}\left({\log\frac{1}{1-t_1}}+{\frac{1}{2}}{\log\frac{1-t_1}{1-t_2}}\right)^m\left({\frac{1}{2}}{\log\frac{1-t_1}{1-t_2}}+{\log\frac{t_2}{t_1}}\right)^n\frac{dt_1dt_2}{(1-t_1)t_2},
\end{eqnarray*}
then
\begin{eqnarray*}
T(m,n)&=&\sum_{a+b=m}\sum_{c+d=n}\left({\frac{1}{2}}\right)^{b+c}{b+c\choose b}\sum_{|\pmb{\alpha}| =b+n+1} \zeta(\{1\}^a,\alpha_1,\ldots,\alpha_{b+c},\alpha_{b+c+1}+1)\\
&=&2{\left(1-\frac{1}{{2}^{m+n+1}}\right)}\zeta(m+n+2)
\end{eqnarray*}
or
\begin{eqnarray*}
T(m,n)&=& {\bf W}(m,n)\zeta(m+n+2)\\
& &+\sum_{j=1}^{\text{min}\{m,n\}}(-1)^j
\sum_{\scriptstyle
{|\pmb{c_j}|=m-j \atop
|\pmb{d_j}|=n-j}}{\zeta(\,c_{j_{0}}+d_{j_{0}}+2, \ldots , c_{j_{j}}+d_{j_{j}}+2\,){\bf W}(c_{j_{0}},d_{j_{0}})}\\
&=&2{\left(1-\frac{1}{{2}^{m+n+1}}\right)}\zeta(m+n+2),
\end{eqnarray*}
where
$${\bf W}(p,q)=\sum_{b=0}^{p}\,\sum_{c=0}^{q}\left({\frac{1}{2}}\right)^{b+c}{b+c\choose b}.$$
|
|---|
