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I would like to find all the possible hypergeometric solutions for the recurrence relation defined as $$ (n+2)a_{n+2} - 2(4n+5)a_{n+1} + 8(2n+1)a_n = 0.$$

Is there any way to approach this problem in an elegant way? As far as I have looked into the book suggested in some other question (A=B), one can use the algorithm hyper to solve these problems. However, I would like a faster way to find all the possible solutions for this concrete recurrence.

Any ideas?

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Excuse me for the newbie question, but what do you mean by hypergeometric solution? I dind't find easily something on google... – Giovanni De Gaetano Jun 27 '11 at 21:10
Can you clarify what you mean by "faster"? Does Hyper really run too slowly on your computer or do you mean that you want a simpler method than the full algorithm because you think that this is a simpler case? – Phira Oct 4 '11 at 10:41

The algorithm Hyper is the general method for solving this kind of recurrence, and there is no reason to believe that this particular case is simple.

It is, of course, possible to guess two answers by inspiration from the source of the problem and prove it by recurrence, but this is hardly a method.

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