On a family of polynomials related to $\zeta(2,1)=\zeta(3)$

TL;DR

This paper provides a new proof of a multiple zeta value identity using generating functions and explores properties of related hypergeometric polynomials, comparing them with conjectural polynomials from prior research.

Contribution

It introduces a novel proof technique for a multiple zeta value identity and analyzes hypergeometric polynomials linked to these values, connecting to conjectural polynomial properties.

Findings

Established a new proof of the zeta value identity using generating functions.

Analyzed hypergeometric polynomials satisfying 3-term recurrence relations.

Compared polynomial properties with those from conjectural identities.

Abstract

We give a new proof of the identity $\zeta(\{2,1\}^l)=\zeta(\{3\}^l)$ of the multiple zeta values, where $l=1,2,\dots$, using generating functions of the underlying generalized polylogarithms. In the course of study we arrive at (hypergeometric) polynomials satisfying 3-term recurrence relations, whose properties we examine and compare with analogous ones of polynomials originated from an (ex-)conjectural identity of Borwein, Bradley and Broadhurst.