scholarly journals EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

2011 ◽  
Vol 07 (03) ◽  
pp. 825-833 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
OLAV K. RICHTER
Keyword(s):  
Fourier Coefficients ◽  
Theta Functions ◽  
Poincaré Series ◽  
Jacobi Forms ◽  
Mock Theta Functions ◽  
Explicit Formulas ◽  
Poincare Series ◽  
Real Analytic

In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.

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.

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.

Generators of Jacobi forms are Poincaré series

2016 ◽  
Vol 45 (3) ◽  
pp. 639-645
Author(s):  
Olav K. Richter ◽  
Howard Skogman
Keyword(s):  
Poincaré Series ◽  
Jacobi Forms ◽  
Poincare Series

scholarly journals NONVANISHING OF JACOBI POINCARÉ SERIES

2010 ◽  
Vol 89 (2) ◽  
pp. 165-179 ◽  
Author(s):  
SOUMYA DAS
Keyword(s):  
Exponential Type ◽  
Poincaré Series ◽  
Jacobi Forms ◽  
Poincare Series ◽  
Integer Weight

AbstractWe prove that, under suitable conditions, a Jacobi Poincaré series of exponential type of integer weight and matrix index does not vanish identically. For the classical Jacobi forms, we construct a basis consisting of the ‘first’ few Poincaré series, and also give conditions, both dependent on and independent of the weight, that ensure the nonvanishing of a classical Jacobi Poincaré series. We also obtain a result on the nonvanishing of a Jacobi Poincaré series when an odd prime divides the index.

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.

scholarly journals On coefficients of Poincaré series and single-valued periods of modular forms

2020 ◽  
Vol 7 (4) ◽  
Author(s):  
Tiago J. Fonseca
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Poincaré Series ◽  
Series Expansions ◽  
Poincare Series ◽  
Integral Weight ◽  
Periods Of Modular Forms

AbstractWe prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$\varGamma _0(N)$$ Γ 0 ( N ) and integral weight $$k\ge 2$$ k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to $$\varGamma _0(N)$$ Γ 0 ( N ) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

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.

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