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.

THE COEFFICIENTS OF THE ω(q) MOCK THETA FUNCTION

2008 ◽  
Vol 04 (06) ◽  
pp. 1027-1042 ◽  
Author(s):  
SHARON ANNE GARTHWAITE
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Theta Functions ◽  
Exact Formula ◽  
Poincaré Series ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Poincare Series ◽  
Vector Valued

In 1920, Ramanujan wrote to Hardy about his discovery of the mock theta functions. In the years since, there has been much work in understanding the transformation properties and asymptotic nature of these functions. Recently, Zwegers proved a relationship between mock theta functions and vector-valued modular forms, and Bringmann and Ono used the theory of Maass forms and Poincaré series to prove a conjecture of Andrews, yielding an exact formula for the coefficients of the f(q) mock theta function. Here we build upon these results, using the theory of vector-valued modular forms and Poincaré series to prove an exact formula for the coefficients of the ω(q) mock theta function.

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’.

scholarly journals Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket

2018 ◽  
Vol 14 (07) ◽  
pp. 1961-1981
Author(s):  
Robert Schneider
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Modular Forms ◽  
Statistical Physics ◽  
Theta Functions ◽  
Triple Product ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Quantum Modular Forms ◽  
Unimodal Sequences

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function [Formula: see text] of Gordon–McIntosh. Here we show that [Formula: see text] arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the [Formula: see text]-bracket of Bloch–Okounkov. Second, we find [Formula: see text] to extend in [Formula: see text] to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other [Formula: see text]-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.

scholarly journals MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS

2013 ◽  
Vol 1 ◽  
Author(s):  
AMANDA FOLSOM ◽  
KEN ONO ◽  
ROBERT C. RHOADES
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Asymptotic Properties ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Radial Limits ◽  
Quantum Modular Forms ◽  
Even Order ◽  
Image Position

AbstractRamanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For the mock theta function $f(q)$, Ramanujan claims that as $q$ approaches an even-order $2k$ root of unity, we have $$\begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )\cdots (1- 2q+ 2{q}^{4} - \cdots )= O(1).\end{eqnarray*}$$ We prove Ramanujan’s claim as a special case of a more general result. The implied constants in Ramanujan’s claim are not mysterious. They arise in Zagier’s theory of ‘quantum modular forms’. We provide explicit closed expressions for these ‘radial limits’ as values of a ‘quantum’ $q$-hypergeometric function which underlies a new relationship between Dyson’s rank mock theta function and the Andrews–Garvan crank modular form. Along these lines, we show that the Rogers–Fine false $\vartheta $-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

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

Harmonic Maass Forms and Mock Modular Forms: Theory and Applications

2017 ◽  
Author(s):  
Kathrin Bringmann ◽  
Amanda Folsom ◽  
Ken Ono ◽  
Larry Rolen
Keyword(s):  
Modular Forms ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Combinatorics of two second-order mock theta functions

2020 ◽  
pp. 1-11
Author(s):  
Hannah Burson
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Second Order ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Third Order ◽  
Combinatorial Interpretations ◽  
Special Cases ◽  
Order Mock Theta Function

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

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].

Sign in / Sign up

Export Citation Format

Share Document