Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations

Gert Almkvista1, Duco van Stratena2 and Wadim Zudilina3

a1 Matematikcentrum, Lunds Universitet, Matematik MNF, Box 118, 22100 Lund, Sweden (

a2 Fachbereich Mathematik 08, Institut für Mathematik, AG Algebraische Geometrie, Johannes Gutenberg-Universität, 55099 Mainz, Germany (

a3 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia (


We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi–Yau differential equations.

(Received July 08 2009)

(Online publication March 30 2011)

Key Words

  • Calabi–Yau differential equation;
  • generalized hypergeometric series;
  • monodromy;
  • algebraic transformation

2010 Mathematics subject classification

  • Primary 33C20; 34M35;
  • Secondary 05A10;
  • 05A19;
  • 14D05;
  • 14J32;
  • 32Q25;
  • 32S40