On an algebraic version of the Knizhnik-Zamolodchikov equation

TL;DR

This paper develops an algebraic difference equation analogue of the Knizhnik-Zamolodchikov equation using Hurwitz polyzeta functions, leading to new insights into multiple Bernoulli polynomials and regularized multiple zeta values.

Contribution

It introduces a difference equation framework for Hurwitz polyzeta functions and establishes algebraic independence and regularization of multiple zeta values at negative integers.

Findings

Normalized multiple Bernoulli polynomials as counterparts to Hurwitz polyzeta functions

Proven algebraic independence of Hurwitz polyzeta functions

Regularization of multiple zeta values at negative integers

Abstract

A difference equation analogue of the Knizhnik-Zamolodchikov equation is exhibited by developing a theory of the generating function $H(z)$ of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which $H(z)$ is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples of negative integers. A generating function for these polynomials satisfies a similar difference equation to that of $H(z)$, but in contrast to the fact that said polynomials have rational coefficients, the algebraic independence of the Hurwitz polyzeta functions is proven. The values of the NMBPs at $z=1$ provide a regularization of the multiple zeta values at tuples of negative integers, which is shown to agree with the regularization given in work of S. Akiyama and collaborators. Various elementary properties of these values are proven.