On the convergence of generalized power series satisfying an algebraic ODE

Abstract

We propose a sufficient condition of the convergence of a generalized power series formally satisfying an algebraic (polynomial) ordinary differential equation. The proof is based on the majorant method.

The Maillet–Malgrange type theorem for generalized power series

“…Note that k = +∞ in Theorem 2 if and only if A m = 0. In this case ϕ does converge in S, which has been already proved in [6] by the majorant method. Here we consider the case k < +∞ using other known methods rather than the majorant one (of course, the convergence could also be proved by these methods).…”

“…As a consequence of the relation (11) and condition that the real parts of the power exponents α, β are positive, we have the following auxiliary lemma (details of the proof see in [6,Lemma 2]).…”

“…where the polynomial L is defined by the formula (6), and exponents α in the monomials z α (δ + s µ ) i of the operator L (z, δ) satisfy the inequalities Re α > 0, Re α (i − p)k.…”

On the Convergence of Formal Exotic Series Solutions of an ODE

“…Thus, our present goal is to obtain some general condition for the convergence of an exotic formal series solution of the equation (1). In this sense, our work continues a series of articles where similar questions were studied for generalized formal power series [4], [6] and formal Dulac series [7], which were inspired by the original paper of B. Malgrange [9] on the classical formal power series solutions of a non-linear ODE. F (x, Φ) = 0, Φ := (ϕ, δϕ, .…”

Towards the Convergence of Generalized Power Series Solutions of Algebraic ODEs

“…The corresponding problem in the differential case was treated by B. Malgrange [16], J. Cano [5] for a formal Taylor series solution, who have proposed sufficient conditions of its convergence and, more generally, estimated the growth of the series coefficients in the case where it diverges (the Maillet-Malgrange theorem). Their results were generalized to formal power series solutions with complex power exponents in [9], [10]. As for the q-difference case, a question of convergence and, more generally, the Maillet-Malgrange type theorem have been studied so far only for formal Taylor series solutions [1,2,7,15,20].…”