Mock Theta Functions Research Articles

Mock modularity and a secondary elliptic genus

The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of some of these invariants. The theory of mock modular forms makes a surprise appearance, shedding light on the integrality properties of some well-known examples.

New mock theta functions and formulas for basic hypergeometric series

AbstractIn recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.

Effective bounds for traces of weak Maass forms

We give an asymptotic formula with an effective bound on the error term for traces of CM values of a generic class of weak Maass forms. We apply this result to give asymptotic formulas with effective bounds on the error terms for the partition function, the Andrews smallest parts function, and the coefficients of Ramanujan's mock theta functions f(q) and ω(q).

RELATIONSHIPS BETWEEN MOCK THETA FUNCTIONS AND q-CONTINUED FRACTIONS

The main object of this paper is to present five new interrelationships between mock theta functions and q-continued fractions.

A single-variable proof of the omega SPT congruence family over powers of 5

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function omega (q). Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 {mathbb {Z}}[X]-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang’s family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of {mathbb {Z}}[X]. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.

An infinite family of vector-valued mock theta functions

An infinite family of vector-valued mock theta functions

Local Maass forms and Eichler–Selberg relations for negative-weight vector-valued mock modular forms

By comparing two different evaluations of a modified (\`{a} la Borcherds) higher Siegel theta lift on even lattices of signature $(r,s)$, we prove Eichler--Selberg type relations for a wide class of negative weight vector-valued mock modular forms. In doing so, we detail several properties of the lift, as well as showing that it produces an infinite family of local (and locally harmonic) Maa{\ss} forms on Grassmanians in certain signatures.

Recurrence relations connecting mock theta functions and restricted partition functions

In this paper, we provide some recurrence relations connecting restricted partition functions and mock theta functions. Elementary manipulations are used including Jacobi triple product identity, Euler’s pentagonal number theorem, and Ramanujan’s theta functions for finding the recurrence relations.

New congruences modulo 9 for the coefficients of Gordon-McIntosh's mock theta function ξ(q)

In recent years, congruence properties for the coefficients of mock theta functions have been studied by mathematicians. Recently, Silva and Sellers proved some congruences modulo 3 and 9 for the coefficients of the third order mock theta function ξ(q) given by Gordon and McIntosh. Motivated by their work, we prove new congruences modulo 9 for the coefficients of ξ(q) based on the formula for the number of representations of an integer as sums of seven squares in this paper. Those congruences involve in_nitely many primes.

Generalizations of mock theta functions and radial limits

In the last letter to Hardy, Ramanujan [Collected Papers, Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by q q -series convergent for | q | > 1 |q|>1 with a complex variable q q , and called these functions “mock theta functions”. Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [A survey of classical mock theta functions, Partitions, q q -series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95–144], they showed that the odd (resp. even) order mock theta functions are related to the function g 3 ( x , q ) g_3(x,q) (resp. g 2 ( x , q ) g_2(x,q) ). These two functions are usually called “universal mock theta functions”. D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109 (2014), pp. 382–422] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell–Lerch sums. In this paper, based on some q q -series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, 1 + ( x q − 1 − x − 1 q ) ∑ n = 0 ∞ ( − 1 ; q ) 2 n q n ( x q − 1 , x − 1 q ; q 2 ) n + 1 = 2 m ( x , q 2 , q ) . \begin{equation*} 1+(xq^{-1}-x^{-1}q)\sum _{n=0}^{\infty }\frac {(-1;q)_{2n}q^{n}}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q). \end{equation*} Then we establish some identities related to these functions and the universal mock theta function g 2 ( x , q ) g_2(x,q) . These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of q q -series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions.

Expressing q-series in terms of building blocks of Hecke-type double-sums

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent [Formula: see text]-hypergeometric function of Andrews as a mixed mock modular form.

Identities involving mock theta functions and theta functions

In this paper, we first present new representations for several mock theta functions. In view of the sign flips on these identities, some interesting results involving theta functions are established by new Bailey pairs and Watson’s 8 ϕ 7 _8\phi _7 transformation formula. Moreover, the truncated forms for the theta series identities are also considered.

Families of identities involving universal mock theta functions

The purpose of this paper is to establish several families of identities involving universal mock theta functions and express them in terms of Appell–Lerch sums. The subsequent consequences of these results are a variety of interesting identities on theta series and classical mock theta functions.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy

We resolve a question of Hikami and Lovejoy that sits at the intersection of quantum modular forms, mock modular forms, and knot theory. We express a family of Hecke–Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. They obtained their family of Hecke–Appell-type sums by considering certain duals of generalized quantum modular forms, where the quantum modular forms are motivated by the colored Jones polynomial for special torus knots.

Graph schemes, graph series, and modularity

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert–Poincaré series of a certain infinite jet scheme. We study new q-representations and examine modular properties of several examples including Dynkin diagrams of finite and affine type. Notably, we obtain new formulas for graph series of type A7 and A8 in terms of “sum of tails” series, and of type D4 and D5 in the form of indefinite theta functions of signature (1,1). We also study examples related to sums of powers of divisors corresponding to 5-cycles. For several examples of graphs, we prove that graph series are mixed quantum modular forms and also mixed mock modular forms.

Some Eichler-Selberg Trace Formulas

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $\Gamma_0(2)$ and $\Gamma_0(4).$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.

New equalities and inequalities for the ranks and cranks of partitions

Let p(n), N(r,m;n) and C(r,m;n) denote the number of partitions of n, the number of partitions of n with rank congruent to r modulo m and the number of partitions of n with crank congruent to r modulo m, respectively. Applying some properties of Appell-Lerch sums and a universal mock theta function g(x,q), we establish the generating functions for N(a,12;n) and C(a,12;n) with 0≤a≤11. With those generating functions, we obtain some new equalities and inequalities involving p(n), N(a,12;n) and C(a,12;n). In particular, we confirm several conjectures due to Aygin and Chan and prove that for n≥0,N(2,12;2n)+N(5,12;2n)+N(6,12;2n)=C(1,12;2n)+C(3,12;2n)+C(5,12;2n).

A domain free of the zeros of the partial theta function

The partial theta function is the sum of the series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|<1$). Its name is due to similaritieswith the formula for the Jacobi theta function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. The function $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently in the study of section-hyperbolic polynomials,i.~e. real polynomials with all coefficients positive,with all roots real negative and all whose sections (i.~e. truncations)are also real-rooted.For each $q$ fixed,$\theta$ is an entire function of order $0$ in the variable~$x$. When$q$ is real and $q\in (0,0.3092\ldots )$, $\theta (q,.)$ is a function of theLaguerre-P\'olyaclass $\mathcal{L-P}I$. More generally, for $q \in (0,1)$, the function $\theta (q,.)$ is the product of a realpolynomialwithout real zeros and a function of the class $\mathcal{L-P}I$. Thus it isan entire function withinfinitely-many negative, with no positive and with finitely-many complexconjugate zeros. The latter are known to belongto an explicitly defined compact domain containing $0$ andindependent of $q$ while the negative zeros tend to infinity as ageometric progression with ratio $1/q$. A similar result is true for$q\in (-1,0)$ when there are also infinitely-many positive zeros.We consider thequestion how close to the origin the zeros of the function $\theta$ can be.In the generalcase when $q$ is complex it is truethat their moduli are always larger than $1/2|q|$. We consider the case when $q$ is real and prove that for any $q\in (0,1)$,the function $\theta (q,.)$ has no zeros on the set $$\displaystyle \{x\in\mathbb{C}\colon |x|\leq 3\} \cap \{x\in\mathbb{C}\colon {\rm Re} x\leq 0\}\cap \{x\in\mathbb{C}\colon |{\rm Im} x|\leq 3/\sqrt{2}\}$$which containsthe closure left unit half-disk and is more than $7$ times larger than it.It is unlikely this result to hold true for the whole of the lefthalf-disk of radius~$3$. Similar domains do not exist for $q\in (0,1)$, Re$x\geq 0$, for$q\in (-1,0)$, Re$x\geq 0$ and for $q\in (-1,0)$, Re$x\leq 0$. We show alsothat for $q\in (0,1)$, the function $\theta (q,.)$ has no real zeros $\geq -5$ (but one can find zeros larger than $-7.51$).