We prove an addition formula for Jacobi functions r ~' ~) (~17_~-~ ) analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structur...

A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over F2. Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broué-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over F2f . We show how to construct Jacobi forms with various ind...

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and repr...

We investigate the action of the heat operator on Jacobi forms. In particular, we present two explicit characterizations of this action on Jacobi forms of index 1. Furthermore, we study congruences and filtrations of Jacobi forms. As an application, we determine when an analog of Atkin’s U-operator applied to a Jacobi form is nonzero modulo a prime.

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at p equally shifted points were recently found. The purpose of this paper is to re-express these cyclic identities in terms of ratios of Jacobi theta functions, since many physicists prefer using Jacobi theta functions rather than Jacobi elliptic functions.

The real-analytic Jacobi forms of Zwegers’ PhD thesis play an important role in the study of mock theta functions and related topics, but have not been part of a rigorous theory yet. In this paper, we introduce harmonic Maass–Jacobi forms, which include the classical Jacobi forms as well as Zwegers’ functions as examples. Maass–Jacobi–Poincaré series also provide prime examples. We compute thei...

The paper contains two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has An coefficients converging to 1, and second, that under an L1-type condition on the Jacobi coefficients, equivalent Jacobi matrices of types 1, 2 and 3 are pairwise asymptotic.

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frölicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity of microcubes.

We use methods from representation theory and invariant theory to compute differential operators invariant under the action of the Jacobi group over a complex quadratic field. This allows us to introduce Maass-Jacobi forms over complex quadratic fields, which are Jacobi forms that are also eigenfunctions of an invariant differential operator. We present explicit examples via Jacobi-Eisenstein s...

We use Jacobi theta functions to construct examples of Jacobi forms over number fields. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function. 2000 Mathemati...