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Partial theta functions and mock modular forms as q-hypergeometric series

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Abstract

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.

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Authors and Affiliations

  1. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Kathrin Bringmann

  2. Department of Mathematics, Yale University, New Haven, CT, 06520-8283, USA

    Amanda Folsom

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Robert C. Rhoades

Authors
  1. Kathrin Bringmann
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  2. Amanda Folsom
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  3. Robert C. Rhoades
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Corresponding author

Correspondence to Amanda Folsom.

Additional information

Dedicated to the visionary Ramanujan, on the 125th anniversary of his birth.

The authors thank Jeremy Lovejoy for several insightful comments. The research of K. Bringmann was supported by the Alfried Krupp Prize for young University Teachers of the Krupp Foundation. A. Folsom is grateful for the support of National Science Foundation grant DMS-1049553. R.C. Rhoades is supported by a NSF Mathematical Sciences Postdoctoral Fellowship.

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Bringmann, K., Folsom, A. & Rhoades, R.C. Partial theta functions and mock modular forms as q-hypergeometric series. Ramanujan J 29, 295–310 (2012). https://doi.org/10.1007/s11139-012-9370-1

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  • DOI: https://doi.org/10.1007/s11139-012-9370-1

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