Recursive Formulas for MacMahon and Ramanujan $q$-series

TL;DR

This paper extends classical work on $q$-series related to divisor sums and partitions, providing explicit formulas and exploring their modular and quasimodular properties, building on MacMahon and Ramanujan's foundational research.

Contribution

It introduces new explicit representations for $q$-series associated with divisor sums and partitions, emphasizing their modular and quasimodular structures.

Findings

Derived explicit formulas for various $q$-series

Analyzed modular and quasimodular properties of these series

Connected classical $q$-series to modern modular form theory

Abstract

In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families of $q$-series by Ramanujan. Our main emphasis will be on explicit representations for a variety of $q$-series, studied primarily by MacMahon and Ramanujan, with an eye towards their modular properties and their proper place in the ring of quasimodular forms of level one and level two.