scholarly journals Partial theta functions and mock modular forms as q-hypergeometric series

2012 ◽  
Vol 29 (1-3) ◽  
pp. 295-310 ◽  
Author(s):  
Kathrin Bringmann ◽  
Amanda Folsom ◽  
Robert C. Rhoades
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Modular Forms ◽  
Partial Theta Functions

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

Asymptotics and Ramanujan's mock theta functions: then and now

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180448
Author(s):  
Amanda Folsom
Keyword(s):  
Modular Forms ◽  
Royal Society ◽  
Theta Functions ◽  
Scientific Meeting ◽  
100Th Anniversary ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Quantum Modular Forms ◽  
Mock Modular Forms ◽  
New Interpretation

This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms—Dyson called this ‘a challenge for the future’ at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann–Rolen, Choi–Lim–Rhoades and Griffin–Ono–Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

2020 ◽  
pp. 1-10
Author(s):  
Amanda Folsom
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Roots Of Unity ◽  
Radial Limit ◽  
Special Values ◽  
Partial Theta Function ◽  
Finite Sums ◽  
Partial Theta Functions

In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity.

Third-order mock theta functions

2019 ◽  
Vol 16 (01) ◽  
pp. 91-106
Author(s):  
Qiuxia Hu ◽  
Hanfei Song ◽  
Zhizheng Zhang
Keyword(s):  
New York ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Dual Nature ◽  
Radial Limits ◽  
The Third ◽  
Lost Notebook

In [G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009), Entry 3.4.7, p. 67; Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24(3) (2011) 345–386; B. Chen, Mock theta functions and Appell–Lerch sums, J. Inequal Appl. 2018(1) (2018) 156; E. Mortenson, Ramanujan’s radial limits and mixed mock modular bilateral [Formula: see text]-hypergeometric series, Proc. Edinb. Math. Soc. 59(3) (2016) 1–13; W. Zudilin, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc. 143(4) (2015) 1471–1476], the authors found the bilateral series for the universal mock theta function [Formula: see text]. In [Choi, 2011], the author presented the bilateral series connected with the odd-order mock theta functions in terms of Appell–Lerch sums. However, the author only derived the associated bilateral series for the fifth-order mock theta functions. The purpose of this paper is to further derive different types of bilateral series for the third-order mock theta functions. As applications, the identities between the two-group bilateral series are obtained and the bilateral series associated to the third-order mock theta functions are in fact modular forms. Then, we consider duals of the second type in terms of Appell–Lerch sums and duals in terms of partial theta functions defined by Hickerson and Mortenson of duals of the second type in terms of Appell–Lerch sums of such bilateral series associated to some third-order mock theta functions that Chen did not discuss in [On the dual nature theory of bilateral series associated to mock theta functions, Int. J. Number Theory 14 (2018) 63–94].

scholarly journals Mock Jacobi forms in basic hypergeometric series

2009 ◽  
Vol 145 (03) ◽  
pp. 553-565 ◽  
Author(s):  
Soon-Yi Kang
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Theta Functions ◽  
Jacobi Forms ◽  
Mock Theta Functions ◽  
Torsion Points ◽  
Weakly Holomorphic Modular Forms ◽  
Real Analytic ◽  
Holomorphic Modular Forms

AbstractWe show that someq-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 modular forms when they are restricted to torsion points and multiplied by suitable powers ofq. We also prove that certain linear sums ofq-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

scholarly journals Three-manifold quantum invariants and mock theta functions

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180439
Author(s):  
Miranda C. N. Cheng ◽  
Francesca Ferrari ◽  
Gabriele Sgroi
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Quantum Invariants ◽  
Quantum Invariant ◽  
Mock Modular Forms ◽  
Mathematical Sciences

Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ (2, 3, 7). This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

SUMS OF PARTIAL THETA FUNCTIONS THROUGH AN EXTENDED BAILEY TRANSFORM

2020 ◽  
pp. 1-10
Author(s):  
MOHAMED EL BACHRAOUI
Keyword(s):  
Hypergeometric Series ◽  
Theta Functions ◽  
Main Tool ◽  
Extended Version ◽  
Bailey Transform ◽  
Partial Theta Functions

In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$ -hypergeometric series.

scholarly journals Quantum mock modular forms arising from eta–theta functions

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Amanda Folsom ◽  
Sharon Garthwaite ◽  
Soon-Yi Kang ◽  
Holly Swisher ◽  
Stephanie Treneer
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Modular Forms
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