Ramanujan-type systems of nonlinear ODEs for [formula omitted] and [formula omitted

Abstract

This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups Γ0(2) and Γ0(3). Using these systems, we provide the generated graded algebras with an sl2(ℂ)-module structure. As applications, we introduce Ramanujan-type tau functions for Γ0(2) and Γ0(3), and obtain some interesting and non-trivial recurrence and congruence relations.