Jacobi form
Introduction In mathematics, a '''Jacobi form''' is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group . The theory  was first systematically studied by .
Definition A Jacobi form of level 1, weight ''k'' and index ''m'' is a function  of two complex variables (with τ in the upper half plane) such that** for all integers λ, μ. * has a Fourier expansion ::
Examples Examples in two variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegel modular forms of genus 2.Examples with more than two variables include characters of some irreducible highest-weight representations of affine Kac–Moody algebras. Meromorphic Jacobi forms appear in the theory of Mock modular forms.
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