Alternating “strange” functions
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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. Here 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 give Cesàro sums for these “strange” series.
Keywordsq-Series Mock theta function Quantum modular form Divergent series
Mathematics Subject Classification33D15 40A30 11F11
The author is thankful to George Andrews for a discussion about divergent series that sparked this study, and to the anonymous referee as well as Olivia Beckwith and my Ph.D. advisor, Ken Ono, for comments that greatly improved the exposition.
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