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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