Compositio Mathematica

Research Article

Mock Jacobi forms in basic hypergeometric series

Soon-Yi Kanga1

a1 Korea Advanced Institute for Science and Technology, Daejeon 305-701, Korea (email:


We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. We also prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

(Received June 11 2008)

(Accepted December 02 2008)

(Online publication April 09 2009)

2000 Mathematics Subject Classification

  • 11F37;
  • 11F50;
  • 05A17;
  • 33D15


  • basic hypergeometric series;
  • mock Jacobi forms;
  • mock modular forms;
  • weakly holomorphic modular forms;
  • ranks and cranks of partitions


This work was supported by a SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government(MEST) (R11-2007-035-01002-0).