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 [math]\displaystyle{ H^{(n,h)}_R }[/math]. The theory was first systematically studied by (Eichler Zagier).
A Jacobi form of level 1, weight k and index m is a function [math]\displaystyle{ \phi(\tau,z) }[/math] of two complex variables (with τ in the upper half plane) such that
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.
Original source: https://en.wikipedia.org/wiki/Jacobi form.
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