Mock Theta Functions Research Articles

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

Congruences between the coefficients of certain mock theta functions

In this article, we prove several congruences modulo 3, 4, 5, 8, 9, 12, 24, 27, 81, 243, and 729 enjoyed by the coefficients of certain mock theta functions. As an example, for the second order mock theta functionsμ2(q):=∑n=0∞(−1)nqn2(q;q2)n(−q2;q2)n2=∑n=0∞Pμ2(n)qn,B2(q):=∑n=0∞qn(n+1)(−q2;q2)n(q;q2)n+12=∑n=0∞qn(−q;q2)n(q;q2)n+1=∑n=0∞PB2(n)qn, we havePμ2(27n+26)≡25PB2(108n+103)(mod27) for all n≥0.

Orbifolded elliptic genera of non-compact models

We revisit the flavored elliptic genus of the N=2 superconformal cigar model and generalize the analysis of the path integral result to the case of real central charge. It gives rise to a non-holomorphic modular covariant function generalizing completed mock modular forms. We also compute the genus for angular orbifolds of the cigar and Liouville theory and decompose it in terms of discrete and continuous contributions. The orbifolded elliptic genus at fractional level is a completed mock modular form with a shadow related to U(1) modular invariants at rational radius squared. We take the limit of the orbifolded genera towards a weighted ground state index and carefully interpret the contributions. We stress that the orbifold cigar and Liouville theories have a maximal and a minimal radius, respectively.

Eulerian series as Hecke-type double sums through the transformation formulas

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of [Formula: see text]-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels [Formula: see text], [Formula: see text], and [Formula: see text].

A Study of Generalized Partial Mock Theta Functions

A Study of Generalized Partial Mock Theta Functions

Generalizations of mock theta functions

Mock theta functions were first introduced by Ramanujan in his last paper to Hardy. Moreover, some other mock theta functions were presented in his lost notebook. It is well known that all the classical mock theta functions can be expressed by the universal mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we establish some generalized mock theta functions and express them in terms of Appell–Lerch sums. Meanwhile, we show that some of Ramanujan’s two-parameter mock theta functions are the special cases of these functions.

Line defect half-indices of SU(N) Chern-Simons theories

We study the Wilson line defect half-indices of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric SU(N) Chern-Simons theories of level k ≤ – N with Neumann boundary conditions for the gauge fields, together with 2d Fermi multiplets and fundamental 3d chiral multiplets to cancel the gauge anomaly. We derive some exact results and also make some conjectures based on expansions of the q-series. We find several interesting connections with special functions known in the literature, including Rogers-Ramanujan functions for which we conjecture integral representations, and the appearance of Appell-Lerch sums for certain Wilson line half-index grand canonical ensembles which reveal an unexpected appearance of mock modular functions. We also find intriguing q-difference equations relating half-indices to Wilson line half-indices. Some of these results also have a description in terms of a dual theory with Dirichlet boundary conditions for the vector multiplet in the dual theory.

Generating functions of the Hurwitz class numbers associated with certain mock theta functions

Generating functions of the Hurwitz class numbers associated with certain mock theta functions

Quantum Modular $\widehat Z{}^G$-Invariants

We study the quantum modular properties of $\widehat Z{}^G$-invariants of closed three-manifolds. Higher depth quantum modular forms are expected to play a central role for general three-manifolds and gauge groups $G$. In particular, we conjecture that for plumbed three-manifolds whose plumbing graphs have $n$ junction nodes with definite signature and for rank $r$ gauge group $G$, that $\widehat Z{}^G$ is related to a quantum modular form of depth $nr$. We prove this for $G={\rm SU}(3)$ and for an infinite class of three-manifolds (weakly negative Seifert with three exceptional fibers). We also investigate the relation between the quantum modularity of $\widehat Z{}^G$-invariants of the same three-manifold with different gauge group $G$. We conjecture a recursive relation among the iterated Eichler integrals relevant for $\widehat Z{}^G$ with $G={\rm SU}(2)$ and ${\rm SU}(3)$, for negative Seifert manifolds with three exceptional fibers. This is reminiscent of the recursive structure among mock modular forms playing the role of Vafa-Witten invariants for ${\rm SU}(N)$. We prove the conjecture when the three-manifold is moreover an integral homological sphere.

On mock theta functions of Gordon and McIntosh

On mock theta functions of Gordon and McIntosh

On Kleinian mock modular forms

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight 2 newforms under the ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}-operator. Our work uses a new non-analytic completion of the Kleinian ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-function from the theory of Abelian functions.

Traces of Poincaré series at square discriminants and Fourier coefficients of mock modular forms

Traces of Poincaré series at square discriminants and Fourier coefficients of mock modular forms

CERTAIN RESULTS INVOLVING q-HYPERGEOMETRIC SERIES AND RAMANUJAN’S MOCK THETA FUNCTIONS

In this paper certain transformation formulas for q-hypergeometric series have been established. In another section of this paper, results involving mock theta functions have also been established.

On Ramanujan's lost notebook and new tenth‐order like identities for second‐, sixth‐, and eighth‐order mock theta functions

AbstractRamanujan'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 identities themselves are of a spectacular nature and were first proved by Choi. We also find eight sixth‐order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth‐orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth‐order mock theta functions that are in the spirit of the six tenth‐order identities. We also include an additional nineteen tenth‐order like identities for various combinations of second‐, sixth‐, and eighth‐order mock theta functions.

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $\omega (q)$ , $\nu (q)$ , and $\psi ^{(3)}(q)$ , introduced by Ramanujan and Watson.

Some Special Integer Partitions Generated by a Family of Functions

In this work, inspired by Ramanujan’s fifth order Mock Theta function f1(q), we define acollection of functions and look at them as generating functions for partitions of some integer n containing at least m parts equal to each one of the numbers from 1 to its greatest part s, with no gaps.We set a two-line matrix representation for these partitions for any m ≥ 2 and collect the values of the sum of the entries in the second line of those matrices. These sums contain information about some parts of the partitions, which lead us to closed formulas for the number of partitions generated by our functions, and partition identities involving other simpler and well known partition functions.

The p-adic constant for mock modular forms associated to CM forms

The p-adic constant for mock modular forms associated to CM forms

A modular approach to Andrews-Beck partition statistics

Andrews recently provided a q-series proof of congruences for NT(m,k,n), the total number of parts in the partitions of n with rank congruent to m modulo k. Motivated by Andrews' works, Chern obtain congruences for Mω(m,k,n) which denotes the total number of ones in the partition of n with crank congruent to m modulo k. In this paper, we focus on the modular approach to these new partition statistics. Applying the theory of mock modular forms, we establish equalities and identities for NT(m,7,n) and Mω(m,7,n).

Generalizations of Mock Theta Functions and Appell–Lerch Sums

Generalizations of Mock Theta Functions and Appell–Lerch Sums

ON CERTAIN RESULTS INVOLVING SQUARE OF RAMANUJAN'S MOCK THETA FUNCTIONS

In this paper, making use of an identity deduced from Bailey's transform, certain results have been established involving the square of Ramanujan's mock theta functions.