Mock Theta Functions Research Articles

Congruences for some partitions related to mock theta functions

Partitions related to mock theta functions were widely studied in the literature. Recently, Andrews et al. introduced two new kinds of partitions counted by [Formula: see text] and [Formula: see text], whose generating functions are [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are two third mock theta functions. Meanwhile, they obtained some congruences for [Formula: see text], [Formula: see text], and the associated smallest parts function [Formula: see text]. Furthermore, Andrews et al. discussed the overpartition analogues of [Formula: see text] and [Formula: see text] which are denoted by [Formula: see text] and [Formula: see text]. In this paper, we derive more congruences for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Moreover, we establish some congruences for [Formula: see text] and its associated smallest parts function [Formula: see text], where [Formula: see text] denotes the number of overpartitions of [Formula: see text] such that all even parts are at most twice the smallest part, and in which the smallest part is always overlined.

Combinatorial interpretations of mock theta functions by attaching weights

This paper provides the combinatorial interpretations of twelve mock theta functions in terms of (n+t)-color partitions with attached weights. We further find combinatorial interpretations of the generalized versions of all these mock theta functions and present these into six families.

A New Approach to Integer Partitions

In this work we define a new set of integer partition, based on a lattice path in $${\mathbb {Z}}^2$$ connecting the line $$x+y=n$$ to the origin, which is determined by the two-line matrix representation given for different sets of partitions of n. The new partitions have only distinct odd parts with some particular restrictions. This process of getting new partitions, which has been called the Path Procedure, is applied to unrestricted partitions, partitions counted by the 1st and 2nd Rogers–Ramanujan Identities, and those generated by the Mock Theta Function $$T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}$$ .

Multiple D3-Instantons and Mock Modular Forms II

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest.

Congruences for partition functions related to mock theta functions

Partitions associated with mock theta functions have received a great deal of attention in the literature. Recently, Choi and Kim derived several partition identities from the third and sixth order mock theta functions. In addition, three Ramanujan-type congruences were established by them. In this paper, we present some new congruences for these partition functions.

Shifted convolution L-series values for elliptic curves

Using explicit constructions of the Weierstrass mock modular form and Eisenstein series coefficients, we obtain closed formulas for the generating functions of values of shifted convolution L-functions associated to certain elliptic curves. These identities provide a surprising relation between weight 2 newforms and shifted convolution L-values when the underlying elliptic curve has modular degree 1 with conductor N such that $$\text {genus}(X_0(N)) = 1$$ .

Properties of certain bilateral mock theta functions-II

Bilateral mock theta functions were obtained and studied in[8]. We express them in terms of Lerch's transcendental function f(x, ξ; q, p). We also express some bilateral mock theta functions as sum of other mock theta functions. We generalize these functions and show that these generalizations are Fq functions. We give an integral representation for these generalized functions.

A unified approach to partial and mock theta functions

A unified approach to partial and mock theta functions

Congruences for a mock modular form on [formula omitted] and the smallest parts function

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL2(Z) modulo primes ℓ≥5. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense.

On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's ω(q)

We analyze the mock modular behavior of P¯ω(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 functions, one of which is ω(q). We find that the modular completion of P¯ω(q) is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.

Representations of superconformal algebras and mock theta functions

It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible $\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invariance occurs for boundary level admissible modules, including when $\frak{g}$ is a basic Lie superalgebra. For example, if $\frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $\hat{\frak{g}}$-modules, whose QHR produces the minimal series modules for the $N=2$ superconformal algebras (resp. a modular invariant family of $N=3$ superconformal algebra modules). However, in the case when $\frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $\hat{\frak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $N=2,3,4$ and big $N=4$ superconformal algebras, whose modified (super)characters span an $SL_2(\mathbf{Z})$-invariant space.

Split (n + t)-color partitions and 2-color F-partitions

Andrews [Generalized Frobenius partitions. Memoirs of the American Math. Soc., 301:1{44, 1984] defined the two classes of generalized F-partitions: F-partitions and k-color F-partitions. For many q-series and Rogers-Ramanujan type identities, the bijections are established between F-partitions and (n + t)-color partitions. Recently (n + t)-color partitions have been extended to split (n+t)-color partitions by Agarwal and Sood [Split (n+t)-color partitions and Gordon-McIntosh eight order mock theta functions. Electron. J. Comb., 21(2):#P2.46, 2014]. The purpose of this paper is to study the k-color F-partitions as a combinatorial tool. The paper includes combinatorial proofs and bijections between split (n + t)-color partitions and 2-color F-partitions for some generalized q-series. Our results further give rise to innate three-way combinatorial identities in conjunction with some Rogers-Ramanujan type identities for some particular cases.

Generalizations of some conjectures of Chan on congruences for Appell–Lerch sums

In this paper, we prove several congruences for Appell–Lerch sums which generalize some conjectures given by Chan.

On the dual nature theory of bilateral series associated to mock theta functions

In recent work, Hickerson and Mortenson introduced a dual notion between Appell–Lerch sums and partial theta functions. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution [Formula: see text] in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell–Lerch sums defined by Mortenson for such functions. Then by using the substitution [Formula: see text] in duals of the second type of each bilateral series, we present how to translate between identities expressing [Formula: see text]-hypergeometric series in terms of Appell–Lerch sums and identities expressing [Formula: see text]-hypergeometric series in terms of partial theta functions. Indeed, we obtain only four duals in terms of partial theta functions of duals of the second type in terms of Appell–Lerch sums of bilateral series associated to mock theta functions. As an application, we construct Ramanujan radial limits by using these bilateral series with mock modular behavior in terms of Appell–Lerch sums for some order mock theta functions. This method is well-suited for the other order mock theta functions.

Black holes and Hurwitz class numbers

We define a natural counting function for BPS black holes in [Formula: see text] compactification of type II string theory, and observe that it is given by a weight 3/2 mock modular form discovered by Zagier. This hints at tantalizing relations connecting black holes, string theory, and number theory.

Ranks of overpartitions modulo 6 and 10

Lovejoy and Osburn proved formulas for the generating functions for the rank differences of overpartitions modulo 3 and 5. In this paper, we derive formulas for the generating functions for the rank differences of overpartitions modulo 6 and 10. With these generating functions, we obtain some equalities and inequalities on ranks of overpartitions modulo 6 and 10. We also relate these generating functions to the third order mock theta functions ω(q) and ρ(q) and the tenth order mock theta functions ϕ(q) and ψ(q).

Hecke structures of weakly holomorphic modular forms and their algebraic properties

Let Sk!(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk!(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=12πi∂∂z. In this paper, we construct a basis of the space Sk!(Γ1(N))/Dk−1(M2−k!(Γ1(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

Small values of coefficients of a half Lerch sum

Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan’s [Formula: see text]-hypergeometric series by relating it to real quadratic field [Formula: see text] and using the arithmetic of [Formula: see text] to solve a conjecture of Andrews on the distributions of its Fourier coefficients. Motivated by Andrews’s conjecture, we discuss an interesting [Formula: see text]-hypergeometric series which comes from a Lerch sum and rank and crank moments for partitions and overpartitions. We give Andrews-like conjectures for its coefficients. We obtain partial results on the distributions of small values of its coefficients toward these conjectures.

New mock theta conjectures Part I

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock $$\theta $$ -functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock $$\theta $$ -functions denoted by $$g_{2}$$ and $$g_{3}$$ . These identities are known as mock $$\theta $$ -conjectures. The fifth- and seventh-order mock $$\theta $$ -conjectures were proved by Dean Hickerson. In the survey paper the authors gave mock $$\theta $$ -conjectures for the other mock $$\theta $$ -functions and referred the proofs to a future paper with this title, listed in their references as [GM4]. The purpose of this paper is to prove these identities for the functions of orders 2 and 3.

Mock modular forms and geometric theta functions for indefinite quadratic forms

Mock modular forms are central objects in the recent discoveries of new instances of Moonshine. In this paper, we discuss the construction of mixed mock modular forms via integrals of theta series associated to indefinite quadratic forms. In particular, in this geometric setting, we realize Zwegers’ mock theta functions of type as line integrals in hyperbolic p-space.