On $N$-Unital Functions

TL;DR

This paper classifies all N-unital rational functions for N up to 4, which are used in studying binomial series and multiple polylogarithms, providing a complete characterization for small N.

Contribution

It determines the full set of N-unital functions for N ≤ 4 using elementary methods and explores some cases at level 5, advancing understanding of their structure.

Findings

Complete classification of N-unital functions for N ≤ 4.

Elementary methods used for classification.

Initial exploration of N=5 case.

Abstract

Let $N$ be a positive integer. We say a non-constant rational function $U(x)\in{\mathbb C}(x)$ is $N$-\emph{unital} if all the zeros and poles of both $U(x)$ and $1-U(x)$ are either 0 or $N$-th roots of unity. These functions are called admissible functions by Au in a recent paper arXiv:2007.03957 and used to study some central binomial series of Ap\'ery type via their iterated integral expression related to multiple polylogrithms and colored multiple zeta values. In this paper we determine the complete set of these functions for $N\le 4$ by elementary method, and briefly study some cases at level 5.