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Families of quasimodular forms arise naturally in many situations such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this note we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two variable generating function for the crank partition statistic is a Jacobi form. Exploiting the structure inherent in the Jacobi theta function we construct explicit expressions for the functions of Atkin and Garvan. Furthermore, this perspective opens the door for further investigation including a study of the moments in arithmetic progressions. We conduct a thorough study of the crank statistic restricted to a residue class modulo 2.
We show that Hidas families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-ad
Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular for
Let $Gamma$ be a finitely generated Fuchsian group of the first kind which has at least one parabolic class. Eichler introduced a cohomology theory for Fuchsian groups, called as Eichler cohomology theory, and established the $CC$-linear isomorphism
Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted to meromorphic Siegel paramodular forms. It was recently observed that the Fourier coefficients of such lifts are then either fast growing or slow growing. In this note we inve
In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Ga