The tenth order mock theta functions revisited

In this paper we consider the first four of the eight identities between the tenth order mock theta functions, found in Ramanujan's lost notebook. These were originally proved by Choi. Here we give an alternative (much shorter) proof.Comment: 11 pages; preprint, submitted for publicatio

Rank-Crank type PDE's for higher level Appell functions

In this paper we consider level l Appell functions, and find a partial differential equation for all odd l. For l=3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l=5 we get a similar PDE found by Garvan.Comment: Preprint, 10 page

On the Fourier coefficients of negative index meromorphic Jacobi forms

In this paper, we consider the Fourier coefficients of meromorphic Jacobi forms of negative index. This extends recent work of Creutzig and the first two authors for the special case of Kac-Wakimoto characters which occur naturally in Lie theory, and yields, as easy corollaries, many important PDEs arising in combinatorics such as the famous rank-crank PDE of Atkin and Garvan. Moreover, we discuss the relation of our results to partial theta functions and quantum modular forms as introducted by Zagier, which together with previous work on positive index meromorphic Jacobi forms illuminates the general structure of the Fourier coefficients of meromorphic Jacobi forms.Comment: 13 pages, minor change

On a completed generating function of locally harmonic Maass forms

While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itelf modular, it can be naturally completed to obtain a half-integral weight modular object

ON THE FOURIER COEFFICIENTS OF POSITIVE INDEX MEROMORPHIC JACOBI FORMS

Fourier coefficients of meromorphic Jacobi forms show up in the study of mock theta functions and Kac-Wakimoto characters. It has previously been shown that they are the holomorphic parts of certain vector-valued almost harmonic Maass forms. In this paper, we give an alternative characterization of these objects by applying the Maass lowering operator to the completions of the Fourier coefficients. We then obtain a formula in terms of classical theta functions and functions that behave like almost holomorphic modular forms

Multivariable Appell functions and nonholomorphic Jacobi forms

Multivariable Appell functions show up in the work of Kac and Wakimoto in the computation of character formulas for certain sl(m, 1) modules. Bringmann and Ono showed that the character formulas for the sl(m, 1) modules L(Lambda((s))), where L(Lambda((s))) is the irreducible sl(m, 1) module with the highest weight Lambda((s)), can be seen as the holomorphic parts of certain nonholomorphic modular functions. Here, we consider more general multivariable Appell functions and relate them to nonholomorphic Jacobi forms