I am not following you reindexing.

The first and fourth sums start with a constants term. The 2nd with term and the third with a

So we need to get all of the series starting at

So we should get something that looks like this

Now collecting constants gives

Now we can each series to start with the and then combine to find the recurrence relations. Also note that That the terms we pulled out can be equated to zero to find .

I hope this gets you started.