A Jacobi theta series and its transformation laws

TL;DR

This paper generalizes Jacobi theta series, showing they are quasi-Jacobi forms, and establishes their transformation laws, proving they are Jacobi forms under certain conditions, with applications to vertex operator algebras.

Contribution

It introduces a generalization of Jacobi theta series, proves their quasi-Jacobi form nature, and establishes transformation laws leading to new Jacobi forms, motivated by vertex operator algebra applications.

Findings

Every generalized Jacobi theta series is a quasi-Jacobi form.

Under certain conditions, these functions satisfy transformation laws as Jacobi forms.

Constructed functions that are also Jacobi forms.

Abstract

We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such functions are Jacobi forms. In establishing these results we construct other functions which are also Jacobi forms. These results are motivated by applications in the theory of vertex operator algebras.