On identities involving the sixth order mock theta functions

We present q-series proofs of four identities involving sixth order mock theta functions from Ramanujan's lost notebook. We also show how Ramanujan's identities can be used to give a quick proof of four sixth order identities of Berndt and Chan

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

An iterative-bijective approach to generalizations of Schur's theorem

We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results, including Schur's theorem, Bressoud's generalization of a theorem of G\"ollnitz, two of Andrews' generalizations of Schur's theorem, and the Andrews-Olsson identities.Comment: 16 page

M_2-rank differences for overpartitions

This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson's conjectures concerning Ramanujan's congruences for the partition function. Since then, other types of rank differences for statistics associated to partitions have been investigated. In this paper, we prove explicit formulas for M_2-rank differences for overpartitions. Additionally, we express a third order mock theta function in terms of rank differences.Comment: 19 page

Torus knots and quantum modular forms

In this paper we compute a $q$-hypergeometric expression for the cyclotomic expansion of the colored Jones polynomial for the left-handed torus knot $(2,2t+1)$ and use this to define a family of quantum modular forms which are dual to the generalized Kontsevich-Zagier series.Comment: 16 page

Overpartition pairs and two classes of basic hypergeometric series

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.Comment: 31 pages, To appear in Adv. Mat

M_2-rank differences for partitions without repeated odd parts

We prove formulas for the generating functions for M_2-rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series.Comment: 18 page

The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, II

Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. The latter case leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. We also use Bailey pairs and formulas of Walsh to find cyclotomic-like expansions for the colored Jones polynomials of $K_{(m,p)}$ and $K_{(m,-p)}$.Comment: 30 pages, to appear in the New York Journal of Mathematic

A partition identity and the universal mock theta function g2g_2

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts satisfy congruence conditions. Furthermore, if small parts are excluded, the resulting overpartitions are generated by the product of a modular form and Gordon and McIntosh's universal mock theta function. Finally, we give an interpretation for the universal mock theta function at real arguments in terms of certain conditional probabilities.Comment: 10 page