Modular forms and q-analogues of modified double zeta values

TL;DR

This paper derives explicit formulas for Hecke eigenforms using q-analogues of modified double zeta values, leading to new period polynomial relations and sum formulas that mirror classical results.

Contribution

It introduces explicit formulas connecting Hecke eigenforms with q-analogues of modified double zeta values, expanding the understanding of their algebraic relations.

Findings

Derived explicit formulas for Hecke eigenforms in terms of q-analogues.

Established period polynomial relations for modified double zeta values.

Produced sum formulas analogous to classical double zeta value identities.

Abstract

We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko and Zagier and the usual sum formulas for classical double zeta values.