The Maillet–Malgrange type theorem for generalized power series

Abstract

There is proposed the Maillet-Malgrange type theorem for a generalized power series (having complex power exponents) formally satisfying an algebraic ordinary differential equation. The theorem describes the growth of the series coefficients. arXiv:1601.05778v1 [math.CA]

On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem

“…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 [6], [7], [8] where similar questions were studied for generalized formal power series…”

“…Proof. Since the polynomial P (λ) = n j=0 a j (0) k + N + iγλ j has no integer roots (in particular, P (−ν k ) = 0) from (7) it follows that ν k is equal to the pole order of ck (t) at t = 0. Hence, as c1 (t) = M(0, t, 0, .…”

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, .…”

“…Let us briefly describe this construction. The functions H, M of the right hand side in (6) are represented by power series convergent in D × ∆:…”

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].…”