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The Ramanujan Journal

, Volume 48, Issue 2, pp 245–250 | Cite as

Alternating “strange” functions

  • Robert SchneiderEmail author
Article
  • 65 Downloads

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 Cesàro sums for these “strange” series.

Keywords

q-Series Mock theta function Quantum modular form Divergent series 

Mathematics Subject Classification

33D15 40A30 11F11 

Notes

Acknowledgements

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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