In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a formal series is an infinite sum that is considered independently from any notion of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
, 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, of the form
where the
called ''coefficients'', are numbers or, more generally, elements of some
ring, and the
are formal powers of the symbol
that is called an
indeterminate or, commonly, a
variable. Hence, power series can be viewed as a generalization of
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s where the number of terms is allowed to be infinite, and differ from usual
power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in
one to one correspondence with their
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.
A formal power series with coefficients in a ring
is called a formal power series over
The formal power series over a ring
form a ring, commonly denoted by
(It can be seen as the
-adic completion of the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
in the same way as the
-adic integers are the -adic completion of the ring of the integers.)
Formal powers series in several indeterminates are defined similarly by replacing the powers of a single indeterminate by
monomials in several indeterminates.
Formal power series are widely used in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
for representing sequences of integers as
generating functions. In this context, a
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
between the elements of a sequence may often be interpreted as a
differential equation that the generating function satisfies. This allows using methods of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
for combinatorial problems (see
analytic combinatorics).
Introduction
A formal power series can be loosely thought of as an object that is like a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
, but with infinitely many terms. Alternatively, for those familiar with
power series (or
Taylor series), one may think of a formal power series as a power series in which we ignore questions of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
by not assuming that the variable ''X'' denotes any numerical value (not even an unknown value). For example, consider the series
If we studied this as a power series, its properties would include, for example, that its
radius of convergence is 1 by the
Cauchy–Hadamard theorem. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s
, −3, 5, −7, 9, −11, ... In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
s
, 1, 2, 6, 24, 120, 720, 5040, ... as coefficients, even though the corresponding power series diverges for any nonzero value of ''X''.
Algebra on formal power series is carried out by simply pretending that the series are polynomials. For example, if
:
then we add ''A'' and ''B'' term by term:
:
We can multiply formal power series, again just by treating them as polynomials (see in particular
Cauchy product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infin ...
):
:
Notice that each coefficient in the product ''AB'' only depends on a ''finite'' number of coefficients of ''A'' and ''B''. For example, the ''X''
5 term is given by
:
For this reason, one may multiply formal power series without worrying about the usual questions of
absolute,
conditional and
uniform convergence which arise in dealing with power series in the setting of
analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.
Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series ''A'' is a formal power series ''C'' such that ''AC'' = 1, provided that such a formal power series exists. It turns out that if ''A'' has a multiplicative inverse, it is unique, and we denote it by ''A''
−1. Now we can define division of formal power series by defining ''B''/''A'' to be the product ''BA''
−1, provided that the inverse of ''A'' exists. For example, one can use the definition of multiplication above to verify the familiar formula
:
An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator