Formal power series

In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).

A formal power series is a special kind of formal series, whose terms are of the form $ax^{n}$ where $x^{n}$ is the $n$ th power of a variable $x$ ( $n$ is a non-negative integer), and $a$ is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the $x^{n}$ are used only as position-holders for the coefficients, so that the coefficient of $x^{5}$ is the sixth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring.

Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to $p$ -adic integers, which can be defined as formal series of the powers of $p$ .

We may give $R^{\mathbb {N} }$ the , where each copy of $R$ is given the discrete topology.

product topology

We may give $R^{\mathbb {N} }$ the , where $I=(X)$ is the ideal generated by $X$ , which consists of all sequences whose first term $a_{0}$ is zero.

I-adic topology

The desired topology could also be derived from the following . The distance between distinct sequences $(a_{n}),(b_{n})\in R^{\mathbb {N} },$ is defined to be $d((a_{n}),(b_{n}))=2^{-k},$ where $k$ is the smallest natural number such that $a_{k}\neq b_{k}$ ; the distance between two equal sequences is of course zero.

metric

Properties[edit]

Algebraic properties of the formal power series ring[edit]

$R[[X]]$ is an associative algebra over $R$ which contains the ring $R[X]$ of polynomials over $R$ ; the polynomials correspond to the sequences which end in zeros.

The Jacobson radical of $R[[X]]$ is the ideal generated by $X$ and the Jacobson radical of $R$ ; this is implied by the element invertibility criterion discussed above.

The maximal ideals of $R[[X]]$ all arise from those in $R$ in the following manner: an ideal $M$ of $R[[X]]$ is maximal if and only if $M\cap R$ is a maximal ideal of $R$ and $M$ is generated as an ideal by $X$ and $M\cap R$ .

Several algebraic properties of $R$ are inherited by $R[[X]]$ :

Generalizations[edit]

Formal Laurent series[edit]

The formal Laurent series over a ring $R$ are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree. That is, they are the series that can be written as

are used to study the properties of multiplicative arithmetic functions

Bell series

are used to define an abstract group law using formal power series

Formal groups

are an extension of formal Laurent series, allowing fractional exponents

Puiseux series

Rational series

Ring of restricted power series

; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007.

Berstel, Jean

: Algebra, IV, §4. Springer-Verlag 1988.

Nicolas Bourbaki

W. Kuich. Semirings and formal power series: Their relevance to formal languages and automata theory. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, Chapter 9, pages 609–677. Springer, Berlin, 1997, 3-540-60420-0

ISBN

Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. :10.1007/978-3-642-01492-5_1

doi

(1990). "Formal Languages and Power Series". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 103–132. ISBN 0-444-88074-7.