I struggle with the following relation that I need (hope) to convert to a closed-form. I write in detail only the point at which I am stuck, as I can take it from there to finish the task. The relation concerns the function $p_{a,b}$, which is a probability, and more specifically we have the following relationships
$$p_{n+a,r}=\left[\prod_{λ=0}^{r-1}γ_1(λ)\right]p_{n+a-r,0},\quad 0\leq r\leq a \tag{1}\label{1}$$
$$p_{n+a,0}=\sum_{κ=0}^{a-1}γ_2(κ)p_{n+a-1,κ} \tag{2}\label{2}$$
From the above I can deduce:
$$p_{n+a,0}=\sum_{κ=0}^{a-1}\left[γ_2(κ)\left[\prod_{λ=0}^{κ-1}γ_1(λ)\right]p_{n+a-κ-1,0}\right] \tag{3}\label{3}$$
The functions $γ_1(\cdot)$ and $γ_2(\cdot)$ are also some (arbitrary) probabilities.
I would like to derive an expression of $p_{n+a,0}$ w.r.t $p_{n,0}$ for all $a\geq 0$. If I do this, then I can derive the closed-form of $p$.