Recursive formula in term of original value

Question

$$P_1=P_0G_{0,1}A_1\\ P_m=P_{m-1}G_{m-1,m}A_m+A_0\sum_{i=0}^{m-2}P_i G_{i,m}~~\text{for}~~m\geq 2$$

Is it possible to write $P_m$ in terms of only $P_0$, i.e., without other $P_j$ terms?

Answer 1

Preliminary questions :

1) are the quantities involved scalars (and not matrices) ?

2) Are the $G_{i,j}$ and the $A_i$ known beforehand ?

If such is the case, I will try my best to cast your general question in a linear algebra setting.

Your 2nd equation(s) can be grouped into a matrix-vector form (may be I am re-discovering the original form of your issue...) :

$$\begin{pmatrix} 1 & -G_{m,m-1}A_m&&\\&1&-G_{m-1,m-2}A_{m-1}&\\ &&\ddots \\ &&&\ddots \end{pmatrix}\begin{pmatrix}P_m\\P_{m-1}\\ \cdots \\ \cdots\end{pmatrix}=A_0 T \begin{pmatrix}P_m\\P_{m-1}\\ \cdots \\ \cdots\end{pmatrix}$$

where $T$ is a lower triangular matrix. Thus, by grouping everything in the LHS for example, you get an equation of the form $AP = 0$; the looked for vector $P$ belongs to the kernel of matrix $A$. The hope, then, is for an explicit knowledge of this kernel.