Consider a sequence of polynomials $Z_N(s,t,p)$ where $N$ is not necessarily the degree and hence not traditional (if its of any use $N$ is the degree only in the case of $N=1$). Suppose we are given a "linear" recurrence relation of the form $Z_N= \sum_{i=1}^{M} w_i(s,t,p)Z_{N-i}$ where $M$ is the order of the recurrence relation. We call this "linear" because we disallow terms such as $Z_{N-1}*Z_{N-2}$ or $Z_{N-3}^2$ and the coefficients are all polynomials.

In the problem I am studying I require a closed form solution for a particular non-traditional 3-variable polynomial recurrence relation of order 9 (i.e. $Z_N$ depends on the previous 9 terms). I'm certain research has been done on the general topic but have had a difficult time finding any. Perhaps I am using the wrong terminology? Any help on that would be appreciated.

Even better of course would be any direct information on these class of problems but of course I'll be happier with a reference as well.

Any info is welcome. Thanks.