On the dual nature theory of bilateral series associated to mock theta functions

2017 ◽  
Vol 14 (01) ◽  
pp. 63-94 ◽  
Author(s):  
Bin Chen
Keyword(s):  
Recent Work ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
The Other ◽  
Mock Theta Functions ◽  
Dual Nature ◽  
Radial Limits ◽  
Dual Notion ◽  
Partial Theta Functions

In recent work, Hickerson and Mortenson introduced a dual notion between Appell–Lerch sums and partial theta functions. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. In this paper, by making the substitution [Formula: see text] in the tail of the associated bilateral series of mock theta functions and universal mock theta functions, we demonstrate how to obtain duals of the second type in terms of Appell–Lerch sums defined by Mortenson for such functions. Then by using the substitution [Formula: see text] in duals of the second type of each bilateral series, we present how to translate between identities expressing [Formula: see text]-hypergeometric series in terms of Appell–Lerch sums and identities expressing [Formula: see text]-hypergeometric series in terms of partial theta functions. Indeed, we obtain only four duals in terms of partial theta functions of duals of the second type in terms of Appell–Lerch sums of bilateral series associated to mock theta functions. As an application, we construct Ramanujan radial limits by using these bilateral series with mock modular behavior in terms of Appell–Lerch sums for some order mock theta functions. This method is well-suited for the other order mock theta functions.

scholarly journals Ramanujan's Radial Limits and Mixed Mock Modular Bilateralq-Hypergeometric Series

2015 ◽  
Vol 59 (3) ◽  
pp. 787-799 ◽  
Author(s):  
Eric Mortenson
Keyword(s):  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Radial Limit ◽  
Dual Nature ◽  
Radial Limits ◽  
Lost Notebook ◽  
Partial Theta Functions ◽  
Limit Result

AbstractUsing results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsomet al.for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateralq-series with mixed mock modular behaviour is well suited for such radial limits. We present five more radial limit results, which follow from mixed mock modular bilateralq-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon and McIntosh. The later bilateral series can be used to compute radial limits for many classical second-, sixth-, eighth- and tenth-order mock theta functions.

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].

ON SOME NEW MOCK THETA FUNCTIONS

2018 ◽  
Vol 107 (1) ◽  
pp. 53-66
Author(s):  
NANCY S. S. GU ◽  
LI-JUN HAO
Keyword(s):  
Theta Functions ◽  
Constant Term ◽  
Second Order ◽  
Sixth Order ◽  
Mock Theta Functions ◽  
Dual Nature ◽  
Partial Theta Functions

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and 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.

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

Bilateral series and Ramanujan radial limits of mock (false) theta functions

2019 ◽  
Vol 30 (04) ◽  
pp. 1950023
Author(s):  
Bin Chen
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Radial Limits ◽  
The Third ◽  
False Theta Functions ◽  
Even Order ◽  
Fifth Order ◽  
Vector Valued

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

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