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On the integrality of power expansions related to hypergeometric series. (English. Russian original) Zbl 1043.11060
Math. Notes 71, No. 5, 604-616 (2002); translation from Mat. Zametki 71, No. 5, 662-676 (2002).
From the 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 power 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 power 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 technics.”

##### MSC:
 11J91 Transcendence theory of other special functions 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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