Heat operators on modular and quasimodular polynomials

Abstract

Jacobi-like forms generalize Jacobi forms, and heat operators on Jacobi forms can be extended to heat operators on Jacobi-like forms. Quasimodular forms correspond to quasimodular polynomials and modular polynomials, and they can be lifted to Jacobi-like forms. We obtain an explicit formula for a differential operator on the space of quasimodular polynomials which is compatible with a heat operators on the space of Jacobi-like forms. We also determine a linear map on the space of modular polynomials that is compatible with this differential operator.