Multiple Zeta Values And Their Applications In Combinatorics

• Main Authors: Chen, Miao, 陳淼
• Other Authors: Eie, Minking
• Format: Others
• Language: en_US
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/69179270410273332462

碩士 === 國立中正大學 === 數學研究所 === 100 === The study of multiple zeta values defined by \begin{align*} \zeta(\alpha)=\sum_{1\leq n_1<n_2<\cdots<n_r}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r} \end{align*} with $\alpha_1$, $\alpha_2$, $\ldots$, $\alpha_r$ positive integers and $\alpha_r\geq2$, had caused a great attention from experts both in physics and mathematics. In this thesis, we produce some combinatorial identities through the shuffle product of multiple zeta values based on the simple fact that the shuffle product of two multiple zeta values of weight $m$ and $n$ will produce $\binom{m+n}{n}$ multiple zeta values of weight $m+n$. In particular, we express the generating functions of some particular products of binomial coefficients in terms of some polynomials. Separating the coefficient then yields combinatorial identities among binomial coefficients. For example, we obtain the following identity \begin{align*} &(j+1)(k-j+1)\binom{k+r+4}{j+\ell+2}\\ &=\sum_{k_1+k_2=k}(k_1+1)(k_2+1)\binom{k_1}{j} \binom{k_2+r+2}{\ell}\\ &\quad\quad+\sum_{k_1+k_2=k}(k_1+1)(k_2+1) \binom{k_1}{k-j} \binom{k_2+r+2}{r-\ell}\\ &\quad\quad+\sum_{k_1+k_2=k}(k_1+1)(k_1+2) \binom{k_1}{j} \binom{k_2+r+1}{\ell}\\ &\quad\quad+\sum_{k_1+k_2=k}(k_1+1)(k_1+2) \binom{k_1}{k-j} \binom{k_2+r+1}{r-\ell}, \end{align*} where $k$, $r$, $j$, $\ell$ are integers with $0\leqslant j \leqslant k$ and $0\leqslant \ell \leqslant r$.