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 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 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.

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 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.

Partial Theta Functions

2009 ◽  
pp. 1-35
Author(s):  
Bruce C. Berndt ◽  
George E. Andrews
Keyword(s):  
Theta Functions ◽  
Partial Theta Functions

Some further Hecke-type identities

2020 ◽  
Vol 16 (09) ◽  
pp. 1945-1967
Author(s):  
Zhizheng Zhang ◽  
Hanfei Song
Keyword(s):  
Series Expansion ◽  
Theta Functions ◽  
Bailey Pairs ◽  
Expansion Formulae ◽  
Partial Theta Functions

In this paper, we obtain some Hecke-type identities by using two [Formula: see text]-series expansion formulae. And, the identities can also be proved directly in terms of Bailey pairs. In particular, we show that certain partial theta functions and the theta functions can be expressed in terms of Hecke-type identities.

On the asymptotics of some partial theta functions

2017 ◽  
Vol 45 (3) ◽  
pp. 895-907 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Theta Functions ◽  
Partial Theta Functions
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