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...

Abstract. In this note, we give some estimates of the generalized quadrature formula of Gauss-Jacobi type for phi-preinvex functions.

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...