In this paper, we consider nonlinear algebraic ordinary differential equations (AODEs) and study their formal power series solutions. Our method is inherited from Lemma 2.2 in [J. Denef and L. Lipshitz, Power series solutions of algebraic differential equations, Mathematische Annalen, 267(1984), 213-238] for expressing high order derivatives of a differential polynomial via their lower order on...

Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this we establish a ...

We give algebraic proofs of transcendence over Q(X) of formal power series with rational coeecients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given.

Some algebraic identities with independent variables are established by means of the calculus on formal power series. Applications to special functions and classical polynomials are also demonstrated.

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when truncated power series can be extended formal solution. If certain regularity condition on the given AODE or initial values is fulfilled, compute all of solutions. Moreover, existence confirmed, present structure set

This paper concerns Laurent series solutions of algebraic ordinary differential equations (AODEs). We first present several approaches to compute formal power series solutions of a given AODE. Then we determine a bound for orders of its Laurent series solutions. Using the order bound, one can transform a given AODE into a new one whose Laurent series solutions are only formal power series. The ...

In this paper, we will take a look at Salem elements dealing with formal power series on Fq, where Fq is finite field . Our main result, presents criteria for an element to be the smallest (SSE) via order extending given in Fq. Moreover, provide CFE of each n.