Iterated integrals, multiple zeta values and Selberg integrals

TL;DR

This paper proves that certain iterated integrals, including those related to Selberg integrals, can be expressed as linear combinations of multiple zeta values, extending known results and providing new connections between integrals and multiple zeta values.

Contribution

The paper reestablishes Brown's theorem on iterated integrals and multiple zeta values, and generalizes Terasoma's result to a broader class of Selberg integrals with coefficients in specific algebraic structures.

Findings

Iterated integrals of a certain form are $Q$-linear combinations of multiple zeta values.

Coefficients of Taylor expansions of Selberg integrals are $Q$-linear combinations of multiple zeta values.

Generalization of Terasoma's result to a wider class of integrals and coefficient functions.

Abstract

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 \[ \mathop{\int\cdots\int}_{0<t_1<\cdots<t_N<1}f\prod_it_i^{\alpha_i}(1-t_i)^{\beta_i}\prod_{i<j}(t_j-t_i)^{\gamma_{ij}} dt_1\cdots dt_N \] (with respect to $\alpha_i,\beta_i,\gamma_{ij}$) 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.