is it possible to obtain a closed-form solution w.r.t. ${P_j:\forall j}$ (or in terms of special functions) for the following equations:
$\frac{\lambda}{\mu}P_0=P_1$
$\frac{\lambda}{\mu}P_j=P_{j+1}+P_{j+2}+\dots+P_{2j+1}$ for $j=1,2,....$
$\sum_{i=0}^\infty P_i=1$
$0<\lambda<\mu$
$P_i>0, \ \forall i=0,1,2,\dots$