# Show $C(B) = C(1) + (1-B)C^*(B)$ where $C(B), C^*(B)$ is are $\infty$-order polynomials

Let $C(B)$ be a $\infty$-order polynomial: $$C(B) = \sum_{k=0}^\infty \alpha_k B^k$$

Show that $$C(B) = C(1) + (1-B)C^*(B)$$ where $C^*(B)$ is a another $\infty$-order polynomial.

This comes from the prove of the Engle-Granger Representation Theorem in their original paper

Here, the polynomials $C(B), C^*(B)$ are moving-average polynomials in time series analysis.

I can't seem to understand how its derived... any help?

To prove this, you need to show every coefficient on the left is equal to the analogous coefficient on the right. To do so, you can use induction, so start with $k=0$, which is trivial, and proceed as you would in the iteration if an induction proof.
Suppose $$1$$ is in the convergent domain of $$C$$. \begin{align} C(B)-C(1) &= \sum_{k=0}^\infty \alpha_k B^k-\sum_{k=0}^\infty \alpha_k \\ &=\sum_{k=0}^\infty \alpha_k (B^k-1) \\ &=\sum_{k=0}^\infty \alpha_k (B-1)\sum_{n=0}^{k-1}B^n \\ &=(B-1)\sum_{k=0}^\infty \sum_{n=0}^{k-1}\alpha_k B^n \\ &=(B-1)\sum_{n=0}^\infty B^n\sum_{k=n+1}^\infty \alpha_k \\ &= (1-B)C^*(B) \end{align} where $$C^*(B)=-\sum_{n=0}^\infty B^n\sum_{k=n+1}^\infty \alpha_k$$ $$\sum_{k=n+1}^\infty \alpha_k$$ exists because $$\sum_{k=0}^\infty \alpha_k$$ converges.