Mock modular grids and Hecke relations for mock modular forms

Abstract

Abstract. We construct and study several infinite grids of mock modular forms. The “rows” of each grid encode an infinite family of forms, while the “columns” encode a second infinite family. Each of these grids contains a generating function of interest as its first entry (for example, the “smallest parts” function of Andrews, or a mock theta function of Ramanujan). We show that the Hecke operators transform these grids in systematic ways, and from this we deduce many corollaries for the generating functions under consideration. For example, we recover the systematic families of congruences for the smallest parts function which have recently been discovered by various authors, and we obtain families of Hecke relations in the grid associated to the mock theta function f(q).