Alternating "strange" functions
Abstract
In this note we consider infinite series similar to the "strange" function $F(q)$ of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-Rhoades, Rolen-Schneider, and others in connection to quantum modular forms. We show that a class of "strange" alternating series that are well-defined almost nowhere in the complex plane can be added (using a modified definition of limits) to familiar infinite products to produce convergent $q$-hypergeometric series, of a shape that specializes to Ramanujan's mock theta function $f(q)$, Zagier's quantum modular form $\sigma(q)$, and other interesting number-theoretic objects. We also discuss Cesàro sums for these alternating series, and continued fractions that are similarly "strange".
- Publication:
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arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.05126
- arXiv:
- arXiv:1701.05126
- Bibcode:
- 2017arXiv170105126S
- Keywords:
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- Mathematics - Number Theory;
- 33D15;
- 40A30
- E-Print:
- 5 pages, updated draft with revised title and exposition, and additional corollaries