Abstract
As known, any formal power series solution φ∈C[[x]] of an algebraic equation is convergent, as well as that of an analytic one. We study the convergence of formal power series solutions of Mahler functional equations F(x,y(x),y(xℓ),…,y(xℓn))=0, where ℓ⩾2 is an integer and F is a holomorphic function near 0∈Cn+2. Extending Bézivin's theorem from the polynomial case to the case under consideration we prove that all such solutions are also convergent. The Newton polygonal method for finding them is explained.
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