The modular differential equation for skew-holomorphic Jacobi forms

In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. On the other hand, this equation differs from the case of holomorphic Jacobi forms introduced by Kiyuna in the types of differential equations and dependences on the index. We show the same properties as previous studies: the solution space of the differential equation is modular invariant and the differential equation is unique.

Automated summary

This paper studies the modular differential equation for skew-holomorphic Jacobi forms, showing similar properties to elliptic modular forms but differing from holomorphic Jacobi forms. The solution space of the differential equation is modular invariant and the equation is unique, as shown in previous studies.