3,427 research outputs found
Partial theta functions and mock modular forms as q-hypergeometric series
Ramanujan studied the analytic properties of many -hypergeometric series.
Of those, mock theta functions have been particularly intriguing, and by work
of Zwegers, we now know how these curious -series fit into the theory of
automorphic forms. The analytic theory of partial theta functions however,
which have -expansions resembling modular theta functions, is not well
understood. Here we consider families of -hypergeometric series which
converge in two disjoint domains. In one domain, we show that these series are
often equal to one another, and define mock theta functions, including the
classical mock theta functions of Ramanujan, as well as certain combinatorial
generating functions, as special cases. In the other domain, we prove that
these series are typically not equal to one another, but instead are related by
partial theta functions.Comment: 13 page
Families of Quasimodular Forms and Jacobi Forms: The Crank Statistic for Partitions
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.Comment: 11 pages. many minor corrections made from previous versio
- …