scholarly journals q-series and weight 3/2 Maass forms

2009 ◽  
Vol 145 (03) ◽  
pp. 541-552 ◽  
Author(s):  
Kathrin Bringmann ◽  
Amanda Folsom ◽  
Ken Ono
Keyword(s):  
Recent Work ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Maass Forms ◽  
Type Series ◽  
Harmonic Maass Forms

AbstractDespite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers–Fine basic hypergeometric series.

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

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.

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.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

2016 ◽  
Vol 152 (6) ◽  
pp. 1159-1197
Author(s):  
Yingkun Li
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Maass Forms ◽  
Stark's Conjecture ◽  
Hilbert Modular ◽  
Harmonic Maass Forms ◽  
The Individual

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

scholarly journals Cycle integrals of meromorphic modular forms and coefficients of harmonic Maass forms

2021 ◽  
Vol 497 (2) ◽  
pp. 124898
Author(s):  
C. Alfes–Neumann ◽  
Kathrin Bringmann ◽  
J. Males ◽  
M. Schwagenscheidt
Keyword(s):  
Modular Forms ◽  
Maass Forms ◽  
Harmonic Maass Forms

Evaluation of some q-integrals in terms of the Dedekind eta function

Analysis ◽  
2018 ◽  
Vol 38 (2) ◽  
pp. 63-79 ◽  
Author(s):  
Greg Doyle ◽  
Kenneth S. Williams
Keyword(s):  
Wide Class ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
General Theorem ◽  
American Mathematical Society ◽  
Basic Hypergeometric Series ◽  
Mathematical Society ◽  
Definite Integral ◽  
Dedekind Eta Function

Abstract A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals.

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