Suppose we have a non-homogeneous linear recurrence relation $$ a(n) + c_{1}a(n-1) + \dots + c_{d}a(n-d) = p(n), \,\, n\in\mathbb{Z} $$ where $c_1,\dots, c_d$ are constants. If $p(n)$ is a polynomial, is there always a particular solution $a(n)$ which is a polynomial?
Basic texts I've looked at are all following a trial-and-error approach, without stating anything general. A pointer to a helpful reference would be much appreciated!
As noted above, there are many resources for this well-studied problem. In particular, when the the RHS is a polynomial of degree $k$ you start with the ansatz $Q(x)$ of the same degree. There are also algorithms. At any rate, here is one online reference in case you find no access to books. http://www.fq.math.ca/Scanned/40-2/el-wahbi.pdf