Poincar\'e and Eisenstein series for Jacobi forms of lattice index

TL;DR

This paper introduces Poincaré and Eisenstein series for Jacobi forms with lattice index, providing their properties, Fourier expansions, and explicit formulas, advancing the understanding of modular forms' structure.

Contribution

It defines Poincaré series for Jacobi forms of lattice index and derives explicit Fourier coefficient formulas, including for the trivial Eisenstein series.

Findings

Fourier expansions of Poincaré and Eisenstein series are computed.

Explicit formulas for Fourier coefficients of trivial Eisenstein series are provided.

Linear combinations of Fourier coefficients of non-trivial Eisenstein series relate to the trivial one.

Abstract

Poincar\'e and Eisenstein series are building blocks for every type of modular forms. We define Poincar\'e series for Jacobi forms of lattice index and state some of their basic properties. We compute the Fourier expansions of Poincar\'e and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one.