I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series:
$$\left(\sum_{n\ge 0}a_nx^n\right)^{-1}=\sum_{n\ge 0}b_nx^n$$
I first thought that my approach would be looking at
$$\frac{1}{\sum_{n\ge 0}a_nx^n}$$
But I'm more thinking that since we know that a series is invertible in the ring if $a_0$ is invertible in the ring of coefficients. Thus, since if we assume it is, and since the unit series is $\{1,0,0,0,….\}$ then we have
$$\left(\sum_{n\ge 0}a_nx^n\right)\left(\sum_{n\ge 0}b_nx^n\right)=1$$
Thus we know that $a_0b_0=1$ and thus $b_0=\frac1{a_0}$. And for the remaining terms we are just looking at the convolution generated by the Cauchy Product and so
$$0=\sum_{j=0}^ka_jb_{k-j}$$
$$-a_0b_k=\sum_{j=1}^ka_jb_{k-j}$$
$$b_k=\frac{-1}{a_0}\sum_{j=1}^ka_jb_{k-j}$$
And thus we have a recursive definition.
Is there another approach that defines the numbers $b_k$ without recursive means? Are you forced to only recursive methods when operating on the ring of formal power series to calculate coefficents?
Best Answer
just a couple of thoughts. the approach you indicate seems more useful and elegant than one based on, say $$ \frac1{1-xP} = 1 +xP+xP^2+\cdots $$ another method might be to use $$ \begin{align} D^1(f^{-1}) &= -f^{-2}f_1 \\ D^2(f^{-1}) &= -f^{-3}(ff_2-2f_1^2) \\ &\cdots \end{align} $$ to build a McLaurin expansion