- 學位論文
Algebraic Relations for Multiple Zeta Values Through Shuffle Product Formulas
Algebraic Relations for Multiple Zeta Values Through Shuffle Product Formulas
並列摘要
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 n, 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 $inom{m+n}{m}$ 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 [ inom{m+n+4}{i+j+2} = sum_{m_{1}+m_{2}=m} rac{inom{m_{1}}{i} inom{m_{2}+n+3}{j+1} + inom{m_{1}}{m-i} inom{m_{2}+n+3}{n-j+1}} ] and egin{align*} inom{i+j}{i} inom{m+n+4}{i+j+2} &= inom{i+j}{i} rac{inom{m+j+3}{i+j+2} + inom{i+n+2}{i+j+2}} \ &quad + sum_{m_{1}+m_{2}=m} sum_{n_{1}+n_{2}=n} inom{n_{1}+i-m_{2}}{n_{1}} inom{n_{1}+m_{1}+2}{m-i+1} inom{m_{2}+j-n_{1}}{m_{2}} inom{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 $j leq n$.