The third order modular linear differential equations

We propose a third order generalization of the Kaneko-Zagier modular differential equation, which has two parameters. We describe modular and quasimodular solutions of integral weight in the case where one of the exponents at infinity is a multiple root of the indicial equation. We also classify solutions of “character type”, which are the ones that are expected to relate to characters of simple modules of vertex operator algebras and one-point functions of two-dimensional conformal field theories. Several connections to generalized hypergeometric series are also discussed.