Related papers: Universal mock theta functions and two-variable He…

In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta…

Multiranks and new rank/crank analogs for a variety of partitions are given, so as to imply combinatorially some arithmetic properties enjoyed by these types of partitions. Our methods are elementary relying entirely on the three classical…

We introduce a statistic on overpartitions called the $\overline{k}$-rank. When there are no overlined parts, this coincides with the $k$-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when…

In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on…

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a…

In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews-Warnaar partial theta function…

We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\pm…

In this paper, we investigate new relationships for bilateral series related to two-parameter mock theta functions, which lead to many identities concerning the bilateral mock theta functions. In addition, interesting relations between the…

We obtain some Bailey pairs associated with indefinite quadratic forms with the $\beta_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo…

We study two aspects of Hecke symmetry in this note: first, we conjecture a generalization of the Ramanujan identities to the case of automorphic forms of Hecke groups; second, we conjecture a generalization of an inversion formula from the…

Mock theta functions were introduced by Ramanujan in 1920 but a proper understanding of mock modularity has emerged only recently with the work of Zwegers in 2002. In these lectures we describe three manifestations of this apparently exotic…

The two partition functions $p_\omega(n)$ and $p_\nu(n)$ were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions $\omega(q)$ and $\nu(q)$, respectively. Recently, Andrews and Yee analytically…

The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of…

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his…

We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic…

We establish some functional identities of theta functions, an elementary proof of classical fourth-order identities, Landen transformations, and q series from the eigenvectors of the discrete Fourier transform. Also, we derive connection…

In 2012 Bryson, Ono, Pitman and Rhoades showed how the generating functions for certain strongly unimodal sequences are related to quantum modular and mock modular forms. They proved some parity results and conjectured some mod 4…

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…

We use a specialization of Ramanujan's ${}_1\psi_1$ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell-Lerch sums and theta functions.