Integrals of logarithmic functions and alternating multiple zeta values

TL;DR

This paper explores explicit relationships between multiple zeta values and integrals of logarithmic functions, deriving recurrence relations and expressing complex zeta values in terms of simpler constants like polylogarithms and ln 2.

Contribution

It introduces new integral representations and recurrence relations for multiple zeta values, enabling their expression in terms of known constants and functions.

Findings

Certain multiple zeta values can be expressed in terms of zeta values, polylogarithms, and ln 2.

Recurrence relations are established for specific classes of multiple zeta values.

Reductions for multiple polylogarithmic values at 1/2 are obtained.

Abstract

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form \[\zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},\bar 1,{{\left\{ 1 \right\}}_{k - 1}}} ),\ (k,m\in \mathbb{N})\] for $m=1$ or $k=1$, and \[\zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},p,{{\left\{ 1 \right\}}_{k - 1}}}),\ (k,m\in\mathbb{N})\] for $p=1$ and $2$, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and $\ln 2$. Moreover, we also prove that the multiple zeta values $\zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},3,{{\left\{ 1 \right\}}_{k - 1}}} )$ can be expressed as a rational linear combination of products of zeta values, multiple polylogarithms and $\ln 2$ when $m=k\in \mathbb{N}$. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at $\frac {1}{2}$.