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The Ramanujan Journal

, Volume 30, Issue 1, pp 67–100 | Cite as

Non-vanishing of Taylor coefficients and Poincaré series

  • Cormac O’Sullivan
  • Morten S. Risager
Article

Abstract

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan’s Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a “generic” point, all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.

Keywords

Modular forms Poincaré series Taylor coefficients CM points 

Mathematics Subject Classification (2000)

11F11 11F25 11F33 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsThe CUNY Graduate CenterNew YorkUSA
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ∅Denmark

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