scholarly journals Duality and Differential Operators for Harmonic Maass Forms

2012 ◽  
pp. 85-106 ◽  
Author(s):  
Kathrin Bringmann ◽  
Ben Kane ◽  
Robert C. Rhoades
Keyword(s):  
Differential Operators ◽  
Maass Forms ◽  
Harmonic Maass Forms

scholarly journals Differential operators on polar harmonic Maass forms and elliptic duality

2019 ◽  
Vol 70 (4) ◽  
pp. 1181-1207
Author(s):  
Kathrin Bringmann ◽  
Paul Jenkins ◽  
Ben Kane
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Poincaré Series ◽  
Maass Forms ◽  
Poincare Series ◽  
Weakly Holomorphic Modular Forms ◽  
Integral Weight ◽  
Harmonic Maass Forms ◽  
Holomorphic Modular Forms

Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

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.

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

Harmonic Maass forms of weight $1$

2015 ◽  
Vol 164 (1) ◽  
pp. 39-113 ◽  
Author(s):  
W. Duke ◽  
Y. Li
Keyword(s):  
Maass Forms ◽  
Harmonic Maass Forms

scholarly journals A classification of harmonic Maass forms

2017 ◽  
Vol 370 (3-4) ◽  
pp. 1729-1758 ◽  
Author(s):  
Kathrin Bringmann ◽  
Stephen Kudla
Keyword(s):  
Maass Forms ◽  
Harmonic Maass Forms

scholarly journals On the Fourier coefficients of meromorphic Jacobi forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1519-1540 ◽  
Author(s):  
René Olivetto
Keyword(s):  
Analytic Functions ◽  
Fourier Coefficients ◽  
Jacobi Form ◽  
Jacobi Forms ◽  
Precise Description ◽  
Maass Forms ◽  
Real Analytic Functions ◽  
Harmonic Maass Forms ◽  
Real Analytic ◽  
Vector Valued

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form φ(z; τ) are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of φ at each pole, as well as some well-known real analytic functions, that appear for instance in the completion of Appell–Lerch sums.

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