Calculating the Fourier Coefficients of Jacobi--Eisenstein series

TL;DR

This paper generalizes Jacobi Eisenstein series to arbitrary lattices, deriving Fourier expansions through two methods, and provides explicit formulas for special lattice cases like even unimodular lattices.

Contribution

It introduces a generalized framework for Jacobi Eisenstein series for arbitrary lattices and derives their Fourier expansions using two distinct methods.

Findings

Fourier expansion formulas for generalized Jacobi Eisenstein series

Explicit formulas for even unimodular lattices

Results applicable to lattices of type NA_{1}

Abstract

In this text we generalize the classical Jacobi Eisenstein series as they were discussed by Eichler and Zagier to arbitrary lattices. We use two different methods to derive the general Fourier expansion. The last two sections give formulas in the case where the lattice is an even unimodular lattice or $L$ is of the type $NA_{1}$.