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

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.

DOI: https://doi.org/10.4213/mzm375

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

Bibliographic databases:

UDC: 511.21+517.588

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
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\paper Integrality of Power Expansions Related to Hypergeometric Series
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\vol 71
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\pages 662--676
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\crossref{https://doi.org/10.4213/mzm375}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1936191}
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\jour Math. Notes
\yr 2002
\vol 71
\issue 5
\pages 604--616
\crossref{https://doi.org/10.1023/A:1015827602930}
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• http://mi.mathnet.ru/eng/mz375
• https://doi.org/10.4213/mzm375
• http://mi.mathnet.ru/eng/mz/v71/i5/p662

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Citing articles on Google Scholar: Russian citations, English citations
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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
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3. Krattenthaler C., Rivoal T., “Analytic Properties of Mirror Maps”, J. Aust. Math. Soc., 92:2 (2012), 195–235
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
5. Delaygue P.E., “Criteria for the Integrality of Taylor Coefficients of Mirror Maps”, J. Reine Angew. Math., 662 (2012), 205–252
6. Delaygue E., “Integrality of the Taylor Coefficients of Roots of Mirror Maps”, J. Theor. Nr. Bordx., 24:3 (2012), 623–638
7. Delaygue E., “A Criterion for the Integrality of the Taylor Coefficients of Mirror Maps in Several Variables”, Adv. Math., 234 (2013), 414–452
8. Roques J., “Arithmetic Properties of Mirror Maps Associated with Gauss Hypergeometric Equations”, Mon.heft. Math., 171:2 (2013), 241–253
9. Roques J., “On Generalized Hypergeometric Equations and Mirror Maps”, Proc. Amer. Math. Soc., 142:9 (2014), 3153–3167
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+
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
12. Fonseca T.J., “Algebraic Independence For Values of Integral Curves”, Algebr. Number Theory, 13:3 (2019), 643–694
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