Getting the closed form solution of a third order recurrence relation with constant coefficients

Question

This is part of the proof of finding the closed from solution of third order recurrence relation

I know that the closed form will look like the following

And this is the part of the proof I can get up to

But what do I need to do after this procedure to get the shape of the closed form mentioned above?

Please can you walk me through the steps?

Answer 1

After you get $B,C,D$, of course, and $g(x)$, then just expand $g(x)$ as a Taylor series. The term $a_n$ is the closed form of recurrence relation you want.