Eichler-Selberg Type Identities for Mixed Mock Modular Forms

TL;DR

This paper develops a framework using holomorphic projection to relate products and Rankin-Cohen brackets of specific mock modular forms and modular forms, leading to new proofs of classical and recent class number relations and extending to mock theta functions.

Contribution

It provides a parametrization of relations among mock modular forms and modular forms, offering new proofs of class number relations and extending the theory to mock theta functions.

Findings

New proofs for classical class number relations

Extension of relations to mock theta functions

A unified parametrization for relations of mock modular forms

Abstract

Using holomorphic projection, we work out a parametrization for all relations of products (resp. Rankin-Cohen brackets) of weight $\tfrac 32$ mock modular forms with holomorphic shadow and weight $\tfrac 12$ modular forms in the spirit of the Kronecker-Hurwitz class number relations. In particular we obtain new proofs for several class number relations among which some are classical, others are relatively new. We also obtain similar results for the mock theta functions.