On the integrality of power expansions related to hypergeometric series. (English. Russian original) Zbl 1043.11060

Zudilin, V. V.

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.”

Reviewer: Tanguy Rivoal (Caen)

MSC:

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

Keywords:

integral power expansion; hypergeometric series; linear differential equation; Calabi-Yau manifold; mirror map