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How do we solve recurrence equations of the form:

$$ax_{n+1}+bx_n+cx_{n-1}=dn^p+e\;,$$ where $a,b,c,d,e$ are constants and $p\in \mathbb Z$?

Perhaps we could first solve the homogeneous equation $$ax_{n+1}+bx_n+cx_{n-1}=0\;.$$ Then we find the particular solution... but how? Guesswork?

Thanks.

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    You can find the particular solution using generating functions. This is closely analogous to using the Laplace transform to find a particular solution to an ODE.2011-12-30
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    This question has been solved perfectly. Hope that the asker has been diving enough and accept the answer at an early date.2012-09-10

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