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Radial limits of mock theta functions

Research in the Mathematical Sciences20152:17

https://doi.org/10.1186/s40687-015-0035-8

  • Received: 6 January 2015
  • Accepted: 25 June 2015
  • Published:

Abstract

Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here, we solve the general problem of understanding Ramanujan’s definition explicitly for any weight \(\frac{1}{2}\) mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.

Keywords

  • Modular Form
  • Theta Function
  • Hypergeometric Series
  • Congruence Subgroup
  • Order Root

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