Mathematical Notes

, Volume 71, Issue 5, pp 604–616

Integrality of Power Expansions Related to Hypergeometric Series

  • V. V. Zudilin

DOI: 10.1023/A:1015827602930

Cite this article as:
Zudilin, V.V. Mathematical Notes (2002) 71: 604. doi:10.1023/A:1015827602930


In the present paper, we study the arithmetic properties of power expansions related to generalized hypergeometric differential equations and series. Defining the series f(z),g(z) in powers of z so that f(z) and \(f\left( z \right)\log z + g\left( z \right)\) satisfy a hypergeometric equation under a special choice of parameters, we prove that the series \(q\left( z \right) = ze^{{{g\left( {Cz} \right)} \mathord{\left/ {\vphantom {{g\left( {Cz} \right)} {f\left( {Cz} \right)}}} \right. \kern-\nulldelimiterspace} {f\left( {Cz} \right)}}}\) in powers of z and its inversion z(q) in powers of q have integer coefficients (here the constant C depends on the parameters of the hypergeometric equation). The existence of an integral expansion z(q) for differential equations of second and third order is a classical result; for orders higher than 3 some partial results were recently established by Lian and Yau. In our proof we generalize the scheme of their arguments by using Dwork's p-adic technique.

integral power expansionhypergeometric serieslinear differential equationCalabi--Yau manifoldmirror map

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. V. Zudilin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityRussia