Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series

TL;DR

This paper connects overpartition M2-rank differences to vector-valued mock Eisenstein series, enabling new class number relations and splitting classical identities into even and odd parts.

Contribution

It establishes a novel link between overpartition rank differences and mock Eisenstein series, leading to new class number formulas and decompositions of classical relations.

Findings

Generated functions relate to mock Eisenstein series.

Derived analogs of class number relations.

Split Kronecker-Hurwitz relation into even and odd parts.

Abstract

We prove that the generating function of overpartition $M2$-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for $M2$-rank differences. As applications we split the Kronecker-Hurwitz relation into its "even" and "odd" parts and calculate sums over Hurwitz class numbers of the form $\sum_{r \in \mathbb{Z}} H(n - 2r^2)$.