Mock Theta Functions Research Articles

Quantum q-series identities

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

New double-sum expansions for certain Mock theta functions

<abstract><p>The study of expansions of certain mock theta functions in special functions theory has a long and quite significant history. Motivated by recent correlations between $ q $-series and mock theta functions, we establish a new $ q $-series transformation formula and derive the double-sum expansions for mock theta functions. As an application, we state new double-sum representations for certain mock theta functions.</p></abstract>

Asymptotics for Bailey-type mock theta functions

We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the circle method due to Wright and a modified Tauberian theorem. We encounter cancelation in our estimates for one of the mock theta functions due to the auxiliary function theta _{n,p} arising from the splitting of Hickerson and Mortenson. We deal with this by using higher-order asymptotic expansions for the Jacobi theta functions.

Three avatars of mock modularity

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 mathematics in three important physical contexts of holography, topology and duality where mock modularity has come to play an important role.

Identities, inequalities and congruences for odd ranks and k-marked odd Durfee symbols

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N0(m,n) denote the number of odd Durfee symbols of n with odd rank m, and N0(r,m;n) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N0(r,12;n) by utilizing some identities involving Appell-Lerch sums m(x,q,z) and a universal mock theta function g(x,q). Based on these formulas, we determine the signs of N0(r,12;4n+t)−N0(s,12;4n+t) for all 0≤r,s≤6 and 0≤t≤3. In particular, we prove that N0(2,12;4n+1)=N0(4,12;4n+1). Moreover, let Dk0(n) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D20(8n+s) and D30(16n+t) are even with s∈{4,6} and t∈{1,9,11,13} which were confirmed by Wang. In this paper, we found new congruences for Dk0(n). In particular, for k=2 or 3, we give characterizations of n such that Dk0(n) is odd and prove that Dk0(n) take even values with probability 1 for n≥0.

Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

Kim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to a pmod {c} for cne 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

Generalizations of the Andrews–Yee identities associated with the mock theta functions $$\omega (q)$$ and $$\nu (q)$$

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third-order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , thereby extending their earlier results with the second author. Generalizing the Andrews–Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang’s three-variable reciprocity theorem. A relation between a new restricted overpartition function $$p^{*}(n)$$ and a weighted partition function $$p_*(n)$$ is obtained from one of the special cases of our second theorem.

Hecke-type triple sums associated with mock theta functions

Hecke-type triple sums associated with mock theta functions

Arithmetic properties for Appell–Lerch sums

Arithmetic properties for Appell–Lerch sums

Some new representations of Hikami’s second-order mock theta function D5(q)

In this paper, a second-order mock theta function $\mathfrak{D}_5(q)$ given by Hikami [11] is studied. By using basic hypergeometric transformation formulae, we attain some new representations of Hikami's mock theta function $\mathfrak{D}_5(q)$. Meanwhile, dual nature of bilateral series associated to mock theta function $\mathfrak{D}_5(q)$ is also discussed.

Integral representations of rank two false theta functions and their modularity properties

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type A_2 and B_2. This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze {hat{Z}}-invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing mathtt{H}-graphs. Along the way, our method clarifies previous results on depth two quantum modularity.

Some new mock theta functions

Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some q-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double sums, we provide the Hecke-type double sums for some third order mock theta functions, and establish the relations between these functions and some classical mock theta functions.

Congruences for the coefficients of the Gordon and McIntosh mock theta function $$\xi (q)$$

Recently Gordon and McIntosh introduced the third order mock theta function \(\xi (q)\) defined by $$\begin{aligned} \xi (q)=1+2\sum _{n=1}^{\infty }\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. \end{aligned}$$Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan-like congruences.

Proofs of Silva–Sellers’ conjectures on a mock theta function

Proofs of Silva–Sellers’ conjectures on a mock theta function

THE ASYMPTOTIC DISTRIBUTION OF TRACES OF WEAK MAASS FORMS

We give an asymptotic formula with a power saving error term for traces of CM values of a generic class of weak Maass forms. We apply this result to study the asymptotic distribution of several arithmetic functions, including the partition function, the Andrews smallest parts function, and the coefficients of Ramanujan's mock theta function f ( q ) .

On the vanishing of some mock theta functions at odd roots of unity

On the vanishing of some mock theta functions at odd roots of unity

Applications of Generalized q-Difference Equations for General q-Polynomials

Andrews gave a remarkable interpretation of the Rogers–Ramanujan identities with the polynomials ρe(N,y,x,q), and it was noted that ρe(∞,−1,1,q) is the generation of the fifth-order mock theta functions. In the present investigation, several interesting types of generating functions for this q-polynomial using q-difference equations is deduced. Besides that, a generalization of Andrew’s result in form of a multilinear generating function for q-polynomials is also given. Moreover, we build a transformation identity involving the q-polynomials and Bailey transformation. As an application, we give some new Hecke-type identities. We observe that most of the parameters involved in our results are symmetric to each other. Our results are shown to be connected with several earlier works related to the field of our present investigation.

Partitions and the Maximal Excludant

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. Let $\sigma\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\sigma\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.

Mocking the u-plane integral

The u-plane integral is the contribution of the Coulomb branch to correlation functions of {mathcal {N}}=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group mathrm{SU}(2), for an arbitrary four-manifold with (b_1,b_2^+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

Parity of coefficients of mock theta functions

We study the parity of coefficients of classical mock theta functions. Suppose g is a formal power series with integer coefficients, and let c(g;n) be the coefficient of qn in its series expansion. We say that g is of parity type (a,1−a) if c(g;n) takes even values with probability a for n≥0. We show that among the 44 classical mock theta functions, 21 of them are of parity type (1,0). We further conjecture that 19 mock theta functions are of parity type (12,12) and 4 functions are of parity type (34,14). We also give characterizations of n such that c(g;n) is odd for the mock theta functions of parity type (1,0).