Hecke-Rogers type series representation on mock theta functions

Abstract

This thesis consists of a collection of results related to Ramanujan’s mock θ-functions, with our primary focus on Hecke-Rogers type series representation on these functions. We provide transformation formulas that lead us to a different representation of the third order mock θ- functions. In a separate section, we give the proofs for three theorems regarding the Hecke-Rogers double series representation associated with definite quadratic forms. In addition, we will provide the Hecke representation for fifth order mock θ-functions, with greater analysis on the last two functions χ0(q) and χ1(q). We will end off this thesis with some examples involving Hecke-Rogers type representation, including the use of these representations to prove Gauss’s famous result that every integer is the sum of three triangular numbers.