How to find general solution of the following difference equation, which follows from a Markov chain?

Question

How to find the general solution of the following difference equation?

$$\alpha(x) = (1-p)\alpha(x-1) + p\alpha(x+2)$$

I constructed the following characteristic equation

$$\lambda = (1-p) + p \lambda^3$$

How to proceed with this?

Answer 1

Ultimately, your question reduces to finding the roots of

$$p\lambda^3 - \lambda + (1-p) = 0$$

From there you can work to establish your solution. Mere observation leads to the finding of one root, $\lambda = 1$. (This is particularly easy to see if you just let $p=1$ since it is - for all intents and purposes in finding that first root - a constant. Of course you still have to account for it in later steps.) Long division or another method of your choice gives the factorization

$$(\lambda - 1)\left(p\lambda^2 + p\lambda + (p-1) \right) = 0$$

Since the new factor is a quadratic, it is easy to proceed from here.