A type of sequence inversion?

Question

I'm reading a textbook which basically has the sequence \begin{align*} a_ n = c_n b_n + \sum_{k=1}^{n-1} w_k b_k \end{align*} and it proceeds to show the inversion of $a_n$ with respect to $b_n$ is \begin{align*} b_n = \frac{1}{c_n} a_n + \sum_{k=1}^{n-1} \tilde{w}_k a_k \end{align*} where $\tilde{w}_n$ are coefficients to be determined that is a function of $w_1, \cdots, w_n, c_n$. The textbook proceeds to find the coefficients via comparison and substitutions. I'm wondering if there is some sort of inversion theorem that would automatically determine these coefficients?

Answer 1

With general $a_{n}, b_{n}$ and $c_{n}$, where $c_{n} \neq 0$, then the best that can be shown is \begin{align} b_{0} &= \frac{a_{0}}{c_{0}} \\ b_{1} &= \frac{a_{1}}{c_{1}} \\ b_{2} &= \frac{a_{2}}{c_{2}} - \frac{w_{1}}{c_{1} \, c_{2}} \, a_{1} \\ b_{3} &= \frac{a_{3}}{c_{3}} - \frac{w_{2}}{c_{2} \, c_{3}} \, a_{2} + \left( \frac{w_{1} w_{2}}{c_{1} c_{2} c_{3}} - \frac{w_{1}}{c_{1} c_{3}} \right) \, a_{1}. \end{align} This takes the form $$b_{n} = \frac{a_{n}}{c_{n}} + \sum_{k=1}^{n-1} \phi_{k}^{n} \, a_{k}.$$

With some relation between $a_{n}$, $b_{n}$, and/or $c_{n}$ then it may be possible to use a generating function, difference operator, etc., to invert the series. Most often it ends up being that a pattern from the first few term is developed.