Quantum Jacobi forms and sums of tails identities
type: Item
instance of: scholarly article
title: Quantum Jacobi forms and sums of tails identities
zbMATH Open document ID: 1491.11051
author: Amanda Folsom
author: Elizabeth Pratt
author: Noah Solomon
author: Andrew R. Tawfeek
published in: Research in Number Theory
publication date: 2022-01-13T00:00:00Z
review text: So-called ``sums of tails'' identities have a long history, going back to Ramanujan, but have seen much recent interest in the modular forms literature due to their connections to mock modular and quantum modular forms. A distinguished sum of tails identity is: \[ \sum_{n\geq0}((q;q)_{\infty}-(q;q)_n)=(q;q)_{\infty}\left(-\frac12+\sum_{n\geq1}\frac{q^n}{1-q^n} \right)+\frac12\sum_{n\geq1}\left(\frac{12}{n}\right)nq^{\frac{n^2-1}{24}}, \] which Zagier used in his pathbreaking study of the Kontsevich function \(\sum_{n\geq0}(q;q)_n\). This allowed Zagier to study its transformation properties, giving an early example of a quantum modular form and in particular to study asymptotics of Vassiliev invariants in knot theory. These quantum modular forms have features similar to modular forms, but instead of being defined on the upper half plane, they are defined on the cusps (namely, on \(\mathbb Q\cup\{i\infty\}\)). To allow for interesting examples, they do not transform strictly as modular forms, but their errors to modularity (differences between two sides in the usual modular symmetry relation) are in some sense ``nice''. This includes properties such as becoming defined on all of \(\mathbb R\), and being smooth, \(\mathcal C^k\), real-analytic, etc. Quantum modular forms were codified in Zagier's seminal 2010 Clay Math Proceedings paper [\textit{D. Zagier}, Clay Math. Proc. 11, 659--675 (2010; Zbl 1294.11084)]. In the years since, this has grown into a large subject with applications throughout math and physics, and many further examples have been uncovered. In [Arch. Math. 107, No. 4, 367--378 (2016; Zbl 1347.05012)], \textit{K. Bringmann} and the first author defined quantum Jacobi forms, which can encode an infinite family of quantum modular forms into a single two-variable function. This has also developed into a topic with many applications and examples in the literature. The current paper is a tour de force of new identities and examples. Firstly, two variable v…
reviewed by: Larry Rolen
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F99
zbMATH DE Number: 7456316
zbMATH Keywords: quantum Jacobi sums
zbMATH Keywords: \(q\)-hypergeometric series
Wikidata QID: Q114218076
MaRDI profile type: Publication
full work available at URL: s40993-021-00304-7
OpenAlex ID: W4200471097
cites work: Ramanujan's ''lost'' notebook. V: Euler's partition identity
cites work: Partitions and indefinite quadratic forms.
cites work: Q3156941
cites work: \(q\)-series identities and values of certain \(L\)-functions
cites work: Quantum Jacobi forms and balanced unimodal sequences
cites work: RANK GENERATING FUNCTIONS FOR ODD-BALANCED UNIMODAL SEQUENCES, QUANTUM JACOBI FORMS, AND MOCK JACOBI FORMS
cites work: Quantum Jacobi forms and finite evaluations of unimodal rank generating functions
cites work: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications
cites work: Partial theta functions and mock modular forms as \(q\)-hypergeometric series
cites work: Half-integral weight Eichler integrals and quantum modular forms
cites work: Universal mock theta functions as quantum Jacobi forms
cites work: q-identities for Maass waveforms
cites work: The theory of Jacobi forms
cites work: Q4040189
cites work: Quantum Jacobi forms in number theory, topology, and mathematical physics
cites work: Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms
cites work: ``Strange'' combinatorial quantum modular forms
cites work: Mock theta functions and quantum modular forms
cites work: A Survey of Classical Mock Theta Functions
cites work: Modular forms and quantum invariants of \(3\)-manifolds
cites work: ON CURIOUS GENERATING FUNCTIONS FOR VALUES OF L-FUNCTIONS
cites work: Q5666618
cites work: Q4040264
cites work: A ``strange'' vector-valued quantum modular form
cites work: On modular forms of half integral weight
cites work: Vassiliev invariants and a strange identity related to the Dedekind eta-function
cites work: Q3001498
cites work: Q3590677
DOI: 10.1007/S40993-021-00304-7
description: scientific article; zbMATH DE number 7456316