Alternating “strange” functions

Abstract

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 Cesaro sums for these “strange” series.