Let us consider a system of recurrence relations such as

$ a_{n-1} = ( lambda_1 + n lambda_2 ) a_n + lambda_3 b_n $

$ b_{n-1} = ( lambda_4 + n lambda_2 ) b_n + lambda_5 a_n $

subject to the initial conditions say, $a_0 = k_1$ and $b_0 = k_2$. So, what I have done so far is that I reexpressed the system as

$v_n = A^T v_{n-1}$

or equivalently,

$ v_n = (A^T)^n v_0 $,

where $ v_n = (a_n, b_n)^T$, and $A$ is the coefficient matrix including $n$.

How do I implement such a system in Mathematica to get the sequences $a_n$ and $b_n$?

Also, is there a function, in Mathematica, similar to RSolve for solving such systems?