论文标题
迭代积分，多个Zeta值和Selberg积分
Iterated integrals, multiple zeta values and Selberg integrals
论文作者
论文摘要
经典的多个Zeta值可以看作是差速器的迭代积分$ \ frac {dt} {t}，\ frac {dt} {1-t} $从$ 0 $到$ 1 $。在本文中，我们谴责布朗定理：对于$ a_i，b_i，c_ {ij} \ in \ mathbb {z} $，表单的迭代积分 \ [ \ Mathop {\ int \ cdots \ int} \ limits_ {0 <t_1 <\ cdots <t_n <1} \ prod_i t_i t_i^{a_i} {a_i}（1-t_i）^{b_i} {b_i} \ prod_} \] 是$ \ mathbb {q} $ - 如果收敛性，权重$ \ leq n $的多个Zeta值的线性组合。 更重要的是，我们表明，如果$ p_i（t），1 \ leq i \ leq n，$在$ \ mathbb {q} \ left [t，1/t，1/t，1/t，1/（1-t）\ right] $ - 代数 - 由多个polylogarithms和他们的双核心和它们的双重元素产生的代数， 如果$ q_ {ij}（t），1 \ leq i <j \ leq n $，则在$ \ mathbb {q} \ left [t，1/t \ right] $ - 对数产生的代数，然后是迭代的积分 \ [ \ Mathop {\ int \ cdots \ int} \ limits_ {0 <t_1 <\ cdots <t_n <1} \ prod_i p_i（t_i）\ prod_ {i <j} q_ {ij} 是$ \ mathbb {q} $ - 多个Zeta值的线性组合。 作为我们主要结果的应用，我们表明Selberg积分的泰勒膨胀系数\ [ \ Mathop {\ int \ cdots \ int} _ {0 <t_1 <\ cdots <t_n <1} f \ prod_it_i^{α_i}（α_i}（1-t_i）^{β_i} \ prod_} dt_1 \ cdots dt_n \]（相对于$α_i，β_i，γ_{ij} $）在某些右半复杂平面的某些产品中的积分处是$ \ mathbb {q} $ - 在\ in \ mathbb {q mathbb {q} q y in \ mathbb {q} Q} Q}的多个Zeta值的线性组合中t_i^{ - 1}，（t_i-t_j）^{ - 1} | 1 \ leq i \ leq n，1 \ leq i <j \ leq n]。$$此语句概括了TeroSoma的原始结果。
Classical multiple zeta values can be viewed as iterated integrals of the differentials $\frac{dt}{t}, \frac{dt}{1-t}$ from $0$ to $1$. In this paper, we reprove Brown's theorem: For $a_i, b_i, c_{ij}\in \mathbb{Z}$, the iterated integral of the form \[ \mathop{\int\cdots \int}\limits_{0<t_1<\cdots<t_N<1}\prod_i t_i^{a_i}(1-t_i)^{b_i} \prod_{i<j}(t_j-t_i)^{c_{ij}}dt_1\cdots dt_N \] is a $\mathbb{Q}$-linear combination of multiple zeta values of weight $\leq N$ if convergent. What is more, we show that if $p_i(t), 1\leq i\leq N, $ are in a $\mathbb{Q}\left[t,1/t, 1/(1-t)\right]$-algebra generated by multiple polylogarithms and their dual, and if $q_{ij}(t), 1\leq i<j\leq N$, are in a $\mathbb{Q}\left[ t,1/t\right]$-algebra generated by logarithm, then the iterated integral \[ \mathop{\int\cdots \int}\limits_{0<t_1<\cdots<t_N<1}\prod_i p_i(t_i)\prod_{i<j}q_{ij}(t_j-t_i)dt_1\cdots dt_N \] is a $\mathbb{Q}$-linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals \[ \mathop{\int\cdots\int}_{0<t_1<\cdots<t_N<1}f\prod_it_i^{α_i}(1-t_i)^{β_i}\prod_{i<j}(t_j-t_i)^{γ_{ij}} dt_1\cdots dt_N \] (with respect to $α_i,β_i,γ_{ij}$) at the integral points in some product of right half complex plane are $\mathbb{Q}$-linear combinations of multiple zeta values for any $$f\in \mathbb{Q}[t_i, t_i^{-1},(t_i-t_j)^{-1}| 1\leq i\leq N, 1\leq i<j\leq N].$$ This statement generalizes Terasoma's original result.
