Hi everyone!

Let $$$R$$$ be a ring, $$$d_0, d_1, d_2, \dots \in R$$$ and $$$e_0, e_1, e_2, \dots \in R$$$ be linear recurrence sequences, such that

In some applications, the following two sequences arise:

Today I'd like to write about the framework that allows to prove that both the sequences defined above are also linear recurrences. It would also allow to compute their characteristic polynomials in $$$O(kl \log kl)$$$, which is optimal as their degrees are $$$O(kl)$$$ in both cases.