Quasimodular forms and modular differential equations which are not apparent at cusps: I

Abstract

In this paper, we explore a two-way connection between quasimodular forms of depth 1 and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases =0+(N) generated by 0(N) and the Atkin-Lehner involutions for N=1,2,3 (0+(1)=SL(2, Z)). Firstly, we note that a quasimodular form of depth 1, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups 0+(N), N=1,2,3.