Vtyjkyuk

I am solving the stationary distirbution of a Discrete time Markov Chain with infinite state space. The state space is $pi^H_0,pi^H_1,...,pi^L_0,pi^L_1,...$. The transition matrix has structure. I eventually reduce the system linear equations into the following two series. But feel no clue about how to start to solve them....
beginalign*
&begincases
&pi^H_0=api^H_1\
&pi^H_1=api^H_0+api^H_2+cpi^L_0\
&pi^H_n=bpi^H_n-1+api^H_n+1,~forall~ngeq 2
endcases~~
begincases
&pi^L_0=cpi^L_1\
&pi^L_1=bpi^H_0+dpi^L_0+cpi^L_2\
&pi^L_n=dpi^L_n-1+cpi^L_n+1,~forall~ngeq 2
endcases\
&sum_n=0^infty(pi^H_n+pi^L_n)=1\
&a+b=1,~c+d=1,~a,b,c,d~mboxare all positive constant pi^H_n_n=0^infty~mboxand pi^L_n_n=0^infty~mboxare all non-negative variables
endalign*

The recurrences in the third lines aren't coupled, so they can be solved with the standard ansatz $pi_k^H=lambda^k$:

$$
lambda^n=blambda^n-1+alambda^n+1;,
$$

which yields the characteristic equation

$$
alambda^2-lambda+b=0
$$

(and likewise for $pi^L$). This yields two linearly independent solutions each, for a total of $4$ free coefficients. The first two lines allow you to express the values for $n=1,2$ in terms of the initial values for $n=0$, yielding one condition each. That leaves two degrees of freedom, and normalization reduces this to one. This might be eliminated if one of the characteristic values has magnitude greater than $1$ and thus isn't suitable for representing probabilities.