Formal Power Series - Interpreting Formal Power Series As Functions

Interpreting Formal Power Series As Functions

In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain. If f = ∑an Xn is an element of R], S is a commutative associative algebra over R, I is an ideal in S such that the I-adic topology on S is complete, and x is an element of I, then we can define

f(X) = \sum_{n\ge 0} a_n X^n.

This latter series is guaranteed to converge in S given the above assumptions on X. Furthermore, we have

and

Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved.

Since the topology on R] is the (X)-adic topology and R] is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients (so that they belong to the ideal (X)): f(0), f(X2−X) and f( (1 − X)−1 − 1) are all well defined for any formal power series f∈R].

With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f whose constant coefficient a = f(0) is invertible in R:

f^{-1} = \sum_{n \ge 0} a^{-n-1} (a-f)^n.

If the formal power series g with g(0) = 0 is given implicitly by the equation

f(g) = X \,

where f is a known power series with f(0) = 0, then the coefficients of g can be explicitly computed using the Lagrange inversion formula.

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