Quantum Jacobi forms and sums of tails identities
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 versions of sums of tails identities are established, which are used to produce quantum Jacobi forms in an analogous way to Zagier's original construction in the one-variable quantum modular case. The authors further establish interesting radial asymptotic identities and provide formulas for such expansions in terms of \(L\)-values. This provides a plethora of new information which recovers many previous examples in the literature, and is a valuable resource for any reader looking to learn more about quantum modular forms or to find useful tools or examples for applications of this subject.
scientific article; zbMATH DE number 7456316
scientific article