ChapterZero

An Inversion Theorem for Formal Power Series

I ran across a very interesting theorem concerning the inversion of formal power series. Recall that if $\{a_n\}$ is a sequence of numbers, then the associated function $A(s) = \sum_{n=0}^\infty a_n s^n$ is called a formal power series; note that A(s) does not need to converge for any particular value of (other than 0)— hence the qualifier ‘formal’. Another name for a formal power series is a generating function, because it is associated with, or generates the sequence $\{a_n\}$ .

Let $A(s) = a_0 + a_1s^1 + a_2s^2 + \ldots$ and $B(s) = b_0 + b_1s^1 + b_2s^2 + \ldots$ be two formal power series and suppose B(0)=b_0 = 0 . Then
$A(B(t)) = a_0 + a_1b_1 t + (a_1b_2 + a_2b_1^2)t^2 + (a_1b_3 + 2a_2b_1b_2 + a_3b_1^3)t^3 + \ldots$ .

Here’s the interesting theorem:

Let a function $B(t) = b_1t + b_2 t^2 + b_3t^3 + \ldots$ be such that and $b_1 \neq 0$ . Then there exist functions
$A(s) = a_1 s + a_2 s^2 + a_3 s^3 + \ldots$ and $C(s) = c_1 s + c_2 s^2 + c_3 s^3 + \ldots$ such that and . Each of the functions and is a unique function possessing this property.

The function is said to be the left inverse of ; is said to be the right inverse. The interesting thing about this theorem is that polynomials are formal power series, so it says that every polynomial has a right and left inverse formal power series. Also, if the polynomial is one like x + x^3 whose range is $\mathcal{R}$ , the left inverse is a power series that converges on all of $\mathcal{R}$ .

My question is, since A(B(t))=t , isn’t A(B(C(t)))=C(t)=A(t) so A=C ?

Possibly relevant posts:

A little about Gröbner bases (6/5/2005)
The Riemann rearrangement theorem, and an interesting corollary (8/14/2006)
The implicit and inverse function theorems (9/12/2006)

An Inversion Theorem for Formal Power Series

Possibly relevant posts:

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