Alternating "strange" functions

TL;DR

This paper explores a class of alternating 'strange' series related to quantum modular forms, showing how they can be combined with infinite products to produce convergent q-hypergeometric series and connect to notable number-theoretic functions.

Contribution

It introduces a modified limit approach to handle 'strange' series, enabling their combination with infinite products to generate meaningful convergent series related to Ramanujan's mock theta functions and quantum modular forms.

Findings

Constructed convergent q-hypergeometric series from 'strange' series and infinite products.

Connected 'strange' series to Ramanujan's mock theta functions and Zagier's quantum modular forms.

Discussed Cesàro summation and continued fractions for these 'strange' series.

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 Ces\`{a}ro sums for these alternating series, and continued fractions that are similarly "strange".