Calculating coefficients of generating functions from a relation.

Question

Given $B(z) = A(z)(1−z) \Leftrightarrow A(z) = B(z)(1 − z)$, derive an inversion formula for the coefficients $a_n$ of $A(z)$, and $b_n$ of $B(z)$.

How do I go about solving this problem?

Answer 1

HINT $$ \begin{split} \sum_{n=0}^\infty a_n z^n &= A(z) = B(z)(1-z) \\ &= B(z) - zB(z) \\ &= \sum_{n=0}^\infty b_n z^n - \sum_{n=0}^\infty b_n z^{n+1}\\ &= \sum_{n=0}^\infty b_n z^n - \sum_{n=1}^\infty b_{n-1} z^n \end{split} $$

Now can you say what $a_n$ looks like in terms of $b_n$? (Be careful with $a_0$).