I'd appreciate your advice on how to solve recurrence relations of the following kind using generating functions:
general solution of:
\begin{cases}a_{n+1} &= 5a_n – 3b_n \\ b_{n+1} &= 4a_n – 2b_n\end{cases}
for $n>0$ and particular solution for $a_0 = 2, b_0 = 1$.
I know how to solve recursions where only one series, f.e. $(a_n)_n$ is involved, but not how to do this for $(a_n)_n$ and $(b_n)_n$ at once.
Best Answer
Note that $$\begin{pmatrix}a_{n+1}\\ b_{n+1}\end{pmatrix}=\begin{pmatrix}5 & -3\\ 4 & -2\end{pmatrix}\begin{pmatrix}a_{n}\\ b_{n}\end{pmatrix}$$
Diagonalize the square matrix to obtain $$A_{n+1}=P^{-1}DPA_n$$ And then $$A_n=P^{-1}D^nPA_0$$