Related papers: On three third order mock theta functions and Heck…

We give a transformation formula for the ``2nd order'' mock theta function which was recently proposed in connection with the quantum invariant for the Seifert manifold.

In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical…

The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular…

We prove certain Nahm-type sum representations for the (odd modulus) Andrews-Gordon identities, the (even modulus) Andrews-Bressoud identities, and Rogers' false theta functions. These identities are motivated on one hand by a recent work…

We find the Hecke-Rogers type series representations of generating functions of the Hurwitz class numbers which is very close to certain mock theta functions. We also prove two combinatorial interpretation of Hurwitz class numbers appeared…

We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions…

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…

We consider the classes of quasimultiplicative, semimultiplicative and Selberg multiplicative functions as extensions of the class of multiplicative functions. We apply these concepts to Ramanujan's sum and its analogue with respect to…

In this paper we obtain explicit formulas for mock theta functions $\Phi^{[m,s]}(\tau, z_1, z_2,t)$ $(m \in \frac12 \mathbf{N}, s \in \frac12 \mathbf{Z})$ by using the coroot lattice of the Lie superalgebra $D(2,1,a)$ and the Kac-Peterson's…

This is the first in a sequence of papers to develop the theory of levels in quantum K-theory and study its applications. Our main results in this paper are toric mirror theorems for permutation-equivariant quantum K-theory with level…

Ramanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth-order mock theta functions and their six identities. The six…

We analyze the mock modular behavior of $\bar{P}_\omega(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta…

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…

In this paper we consider the first four of the eight identities between the tenth order mock theta functions, found in Ramanujan's lost notebook. These were originally proved by Choi. Here we give an alternative (much shorter) proof.

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in…

Recently, Nath and Das investigated congruence properties for the second order mock theta function $B(q)$. In their paper, they asked for analytic proofs of three identities on the second order mock theta functions $A(q)$, $B(q)$ and…

Ramanujan derived a sequence of even weight $2n$ quasimodular forms $U_{2n}(q)$ from derivatives of Jacobi's weight $3/2$ theta function. Using the generating function for this sequence, one can construct sequences of quasimodular forms of…

In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity.…

Using Appell function properties we give short proofs of Ramanujan-like identities for the eighth order mock theta function $V_0(q)$ after work of Chan and Mao; Mao; and Brietzke, da Silva, and Sellars. We also present a generalization of…