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Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

For example, I might want to approximate

$$ a_{n+k}=a_{n+k-1}+a_{n+k-2}+...+a_n $$

with a geometric series

$$ b_{n+1}=qb_n $$

using some standard method.

I'd like to estimate the series when the root of the characteristic equation is difficult to find or doesn't have an analytic solution.

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