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Mat. Zametki, 2002, Volume 71, Issue 5, Pages 662–676 (Mi mz375)  

This article is cited in 12 scientific papers (total in 12 papers)

Integrality of Power Expansions Related to Hypergeometric Series

W. V. Zudilin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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 $f(z)\log z+g(z)$ satisfy a hypergeometric equation under a special choice of parameters, we prove that the series $q(z)=ze^{g(Cz)/f(Cz)}$ 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.


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English version:
Mathematical Notes, 2002, 71:5, 604–616

Bibliographic databases:

UDC: 511.21+517.588
Received: 31.10.2000

Citation: W. V. Zudilin, “Integrality of Power Expansions Related to Hypergeometric Series”, Mat. Zametki, 71:5 (2002), 662–676; Math. Notes, 71:5 (2002), 604–616

Citation in format AMSBIB
\by W.~V.~Zudilin
\paper Integrality of Power Expansions Related to Hypergeometric Series
\jour Mat. Zametki
\yr 2002
\vol 71
\issue 5
\pages 662--676
\jour Math. Notes
\yr 2002
\vol 71
\issue 5
\pages 604--616

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    This publication is cited in the following articles:
    1. Krattenthaler Ch., Rivoal T., “On the Integrality of the Taylor Coefficients of Mirror Maps, II”, Commun. Number Theory Phys., 3:3 (2009), 555–591  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    2. Krattenthaler Ch., Rivoal T., “ON THE INTEGRALITY OF THE TAYLOR COEFFICIENTS OF MIRROR MAPS”, Duke Mathematical Journal, 151:2 (2010), 175–218  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    3. Krattenthaler C., Rivoal T., “Analytic Properties of Mirror Maps”, J. Aust. Math. Soc., 92:2 (2012), 195–235  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. Lau S.-Ch., Leung N.C., Wu B., “Mirror Maps Equal SYZ Maps for Toric Calabi-Yau Surfaces”, Bull. London Math. Soc., 44:Part 2 (2012), 255–270  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Delaygue P.E., “Criteria for the Integrality of Taylor Coefficients of Mirror Maps”, J. Reine Angew. Math., 662 (2012), 205–252  mathscinet  zmath  isi  elib
    6. Delaygue E., “Integrality of the Taylor Coefficients of Roots of Mirror Maps”, J. Theor. Nr. Bordx., 24:3 (2012), 623–638  crossref  mathscinet  zmath  isi  scopus  scopus
    7. Delaygue E., “A Criterion for the Integrality of the Taylor Coefficients of Mirror Maps in Several Variables”, Adv. Math., 234 (2013), 414–452  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. Roques J., “Arithmetic Properties of Mirror Maps Associated with Gauss Hypergeometric Equations”, Mon.heft. Math., 171:2 (2013), 241–253  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Roques J., “On Generalized Hypergeometric Equations and Mirror Maps”, Proc. Amer. Math. Soc., 142:9 (2014), 3153–3167  crossref  mathscinet  zmath  isi  scopus  scopus
    10. Delaygue E., Rivoal T., Roques J., “On Dwork?s -adic formal congruences theorem and hypergeometric mirror maps”, Mem. Am. Math. Soc., 246:1163 (2017), 1+  crossref  mathscinet  isi  scopus
    11. Cho Ch.-H., Hong H., Lau S.-Ch., “Localized Mirror Functor For Lagrangian Immersions, and Homological Mirror Symmetry For P-a,B,C(1)”, J. Differ. Geom., 106:1 (2017), 45–126  crossref  mathscinet  zmath  isi  scopus
    12. Fonseca T.J., “Algebraic Independence For Values of Integral Curves”, Algebr. Number Theory, 13:3 (2019), 643–694  crossref  mathscinet  zmath  isi
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