I want to simplify the following expression for $q_i$ and $p_i$, where $i \in \{1,2,...,n \}$ and $q_i \in [0,1]$ and $p_i \in [0,1] $ , is there any standard method for it? For example, based on generating functions or ... .
The expression is
$$ \displaystyle\frac{1}{1-\displaystyle\frac{q_1 p_1}{1-\displaystyle\frac{q_2 p_2}{...}}}$$
$$\frac1{1-\dfrac{t_1}{1-\dfrac{t_2}{1-\dfrac{t_3}{1-t_4}}}}=\frac1{1-\dfrac{t_1}{1-\dfrac{t_2(1-t_4)}{1-t_4-t_3}}}=\frac1{1-\dfrac{t_1(1-t_4-t_3)}{1-t_4-t_3-t_2(1-t_4)}} \\=\dfrac{1-t_4-t_3-t_2(1-t_4)}{1-t_4-t_3-t_2(1-t_4)-t_1(1-t_4-t_3)} \\=\dfrac{1-t_4-t_3-t_2+t_2t_4}{1-t_4-t_3-t_2-t_1+t_2t_4+t_1t_4+t_1t_3}.$$
With more levels, the expressions will just be polynomials of higher degree, and no simplification is possible.