with coefficients ak ∈ C (k ∈ Z ) are important in many branches of mathematics. Maple supports the computation of truncated series with its series command, and through the powerseries package [6] infinite series are available. In the latter case, the series is represented as a table of coefficients that have already been determined together with a function for computing additional coefficients...

The set S[[x]] of formal power series in x over a set S is the set of functions from the nonnegative integers to S. However, the way that we represent elements of S[[x]] will be as an infinite series, and operations in S[[x]] will be closely linked to the addition and multiplication of finite-degree polynomials. This paper will introduce a bit of the structure of sets of formal power series and...

Quotient is a basic operation of formal languages, which plays key role in the construction minimal deterministic finite automata (DFA) and universal automata. In this paper, we extend to power series systemically investigate its implications study weighted particular, define two quotient operations for are called (left or right) quotients residuals respectively. Algebraic properties closure (w...

We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributi...

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best known algorithm for computing the Nth coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in √ N . We show that in positive characteristic p, the complexity can be lowered to O(logN). The mathematical basis of this dramatic i...