Mock modular grids and Hecke relations for mock modular forms

Scott Ahlgren, Byungchan Kim

Research output: Contribution to journalArticlepeer-review

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).

Original languageEnglish (US)
Pages (from-to)1261-1287
Number of pages27
JournalForum Mathematicum
Volume26
Issue number4
DOIs
StatePublished - Jul 1 2014

Keywords

  • Hecke relation
  • Mock modular form
  • Mock modular grid
  • Mock theta function
  • Smallest parts function

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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