Quantum mock modular forms arising from eta–theta functions

@article{Folsom2016QuantumMM,
  title={Quantum mock modular forms arising from eta–theta functions},
  author={Amanda Folsom and Sharon Anne Garthwaite and Soon-Yi Kang and Holly Swisher and Stephanie Treneer},
  journal={Research in Number Theory},
  year={2016},
  volume={2},
  pages={1-41}
}
In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta–theta functions by constructing mock modular forms from the eta–theta functions with even characters, such that the shadows of these mock modular forms are given by the eta–theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate… 
Extending a catalog of mock and quantum modular forms to an infinite class
Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed
Quantum modularity of mock theta functions of order 2
In [9], we computed shadows of the second order mock theta functions and showed that they are essentially same with the shadow of a mock theta function related to the Mathieu moonshine phenomenon. In
MOCK THETA FUNCTIONS OF ORDER 2 AND THEIR SHADOW COMPUTATIONS
Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular form of weight 1/2 by adding a period integral of a certain weight 3/2 unary theta series. This
Quantum Modular Forms and Singular Combinatorial Series with Distinct Roots of Unity
Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation
Quantum modular forms and singular combinatorial series with repeated roots of unity
In 2007, G.E. Andrews introduced the (n+1)-variable combinatorial generating function Rn(x1, x2, · · · , xn; q) for ranks of n-marked Durfee symbols, an (n+1)-dimensional multisum, as a vast
Universal mock theta functions as quantum Jacobi forms
Quantum Jacobi forms were defined in 2016, naturally combining Zagier’s definition of a quantum modular form with that of a Jacobi form. To date, just three examples of such functions exist in the
Quantum Jacobi forms in number theory, topology, and mathematical physics
  • A. Folsom
  • Mathematics, Physics
    Research in the Mathematical Sciences
  • 2019
We establish three infinite families of quantum Jacobi forms, arising in the diverse areas of number theory, topology, and mathematical physics, and unified by partial Jacobi theta functions.
RANK GENERATING FUNCTIONS FOR ODD-BALANCED UNIMODAL SEQUENCES, QUANTUM JACOBI FORMS, AND MOCK JACOBI FORMS
Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to
Transport in Unsteady Flows: from Deterministic Structures to Stochastic Models and Back Again (17w5048)
Infinite-Dimensional Systems Infinite-dimensional system theory is an area where the goal is to axiomatize certain properties in control theory within the context of infinite-dimensional theory with
Women in Numbers 4
The first Women in Numbers workshop was held at BIRS in 2008, with the explicit goals of increasing the participation of women in number theory research and highlighting the contributions of women

References

SHOWING 1-10 OF 41 REFERENCES
Unimodal sequences and quantum and mock modular forms
TLDR
It is shown that the rank generating function U(t; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms, and a new representation is obtained for a certain generating function for L-values.
Hecke‐type double sums, Appell–Lerch sums, and mock theta functions, I
By introducing a dual notion between partial theta functions and Appell–Lerch sums, we find and prove a formula which expresses Hecke‐type double sums in terms of Appell–Lerch sums. Not only does our
A “strange” vector-valued quantum modular form
Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions
Eta-quotients and theta functions
Mock Theta Functions
The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider
Mock Jacobi forms in basic hypergeometric series
Abstract We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic
A Survey of Classical Mock Theta Functions
In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an
Half-integral weight Eichler integrals and quantum modular forms
On the seventh order mock theta functions
In a recent paper [H], we proved the "Mock Theta Conjectures". These are identities, stated by Ramanujan in his "lost notebook" JR2, pp. 19-20], involving two of the 5th order mock 0-functions. In
...
...