Ramanujan-type systems of nonlinear ODEs for ${\Gamma }_0(2)$ and ${\Gamma }_0(3)$

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 ${\Gamma }_0(2)$ and ${\Gamma }_0(3)$. Using these systems, we provide the generated graded algebras with an ${\mathfrak{sl}}_2(\mathbb{C})$-module structure. As applications, we introduce Ramanujan-type tau functions for ${\Gamma }_0(2)$ and ${\Gamma }_0(3)$, and obtain some interesting and non-trivial recurrence and congruence relations.

21 pages