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:

$\alpha P_0=P_1$, $\alpha<1$

$\alpha P_j=P_{j+1}+P_{j+2}+\dots+P_{2j+1}$ for $j=1,2,....$

$\sum_{i=1}^\infty P_i=1$

$P_i\geq 0, \forall i$

OR let me put the very original equations below:

$\lambda P_0=\mu P_1$

$\lambda P_{j-1} + \mu (P_{2j}+P_{2j+1})=(\lambda+\mu)P_j, \forall j>0$

$\sum_{i=1}^\infty P_i=1$

$P_i\geq 0, \forall i$