An identity for generating series of deformations of multiple zeta values within an algebraic framework

TL;DR

This paper generalizes an identity relating generating series of deformations of multiple zeta values within an algebraic framework, connecting them to Eisenstein series and periodlike functions, and introduces a new formula for these series.

Contribution

It extends Bachmann's identity to a broader algebraic setting and derives a new formula involving periodlike functions for deformations of multiple zeta values.

Findings

Generalized Bachmann's identity within an algebraic framework

Derived a new formula involving periodlike functions

Connected deformations of multiple zeta values to Eisenstein series

Abstract

Bachmann proves an identity expressing the generating series of MacMahon's generalized sum-of-divisors $q$-series in terms of Eisenstein series. MacMahon's $q$-series can be regarded as a $q$-analogue of the multiple zeta value $\zeta(2, 2, \ldots , 2)$, up to a power of $1-q$. Based on this observation, we generalize Bachmann's identity within an algebraic framework and prove a general identity. As a byproduct, we obtain a formula for the generating series of another deformation of multiple zeta values defined by the author. In this formula, periodlike functions introduced by Lewis and Zagier appear as a counterpart of Eisenstein series.