Taylor coefficients of non-holomorphic Jacobi forms and applications

In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of holomorphic Jacobi forms are quasimoular forms. However recently there has been a wide interest for Taylor coefficients of non-holomorphic Jacobi forms for example arising in combinatorics. In this paper, we show that such coefficients still inherit modular properties. We then work out the precise spaces in which these coefficients lie for two examples

Improved bounds for Fourier coefficients of Siegel modular forms

The goal of this paper is to improve existing bounds for Fourier coefficients of higher genus Siegel modular forms of small weight

On the explicit construction of higher deformations of partition statistics

The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for $p(n)$. In a series of papers the author and Ono \cite{BO1,BO2} connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms and which were first considered in \cite{BF}. Here we do a further step towards understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews \cite{An1} into the framework of weak Maass forms. To do this we construct a new class of functions which we call quasiweak Maass forms because they have quasimodular forms as components. As an application we prove two conjectures of Andrews. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves, and also their derivatives. As a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices.Comment: 29 pages, Duke J. accepted for publicatio

Regularized inner products and weakly holomorphic Hecke eigenforms

We show that the image of repeated differentiation on weak cusp forms is precisely the subspace which is orthogonal to the space of weakly holomorphic modular forms. This gives a new interpretation of the weakly holomorphic Hecke eigenforms

A Framework for Modular Properties of False Theta Functions

False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the Circle Method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.Comment: 20 page

Dyson's Rank, overpartitions, and weak Maass forms

In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions.Comment: 24 pages IMRN, accepted for publicatio

Polar harmonic Maass forms and their applications

In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points

Asymptotic formulas for stacks and unimodal sequences

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions. These results describe combinatorial properties, generating functions, and asymptotic formulas for the enumeration functions. We also prove several new asymptotic results that fill in the notable missing cases from the literature, including an open problem in statistical mechanics due to Temperley. Furthermore, we explain the combinatorial and asymptotic relationship between partitions, Andrews' Frobenius symbols, and stacks with summits.Comment: 19 pages, 4 figure