Formal Power Series - Introduction

Introduction

A formal power series can be loosely thought of as an object that is like a polynomial, 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 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. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of coefficients . 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 factorials as coefficients, even though the corresponding power series diverges for any nonzero value of X.

Arithmetic 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):

Notice that each coefficient in the product AB only depends on a finite number of coefficients of A and B. For example, the X5 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.

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 B A −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 for a formal power series in one variable extracts the coefficient of Xn, and is written e.g. A, so that A = 5 and A = −11. Other examples include

and

Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.