Curious sequences of polynomials

Question

Given an integer $k\geq 2$, and $k+1$ invertible initial values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$ we set $$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{n-j}} {s_{n-k}}$$ as long as $s_0,s_1,\ldots,s_{n-k}$ are invertible in $\mathcal A(q)$.

We get seemingly $s_n\in \mathcal A[q]$ with coefficients in $\mathcal A$ up to denominators which are products of $s_0,\ldots,s_k$. (It is of course also possible to choose $s_0,\ldots,s_k$ in $ \mathcal A[q]$ or in $\mathcal A(q)$. Denominators of $s_n$ involve seemingly also only factors occuring in $s_0,\ldots,s_k$.)

In particular, if $s_0,\ldots,s_k$ are all in $\{\pm 1\}$, we get seemingly integral polynomials $s_n$ in $\mathbb Z[q]$.

Has anybody a clue why this could be true?

Complements: For $s_0=s_1=\ldots=s_k=1$ the evaluation at $q=0$ yields integral sequences related to solutions of so-called Hurwitz equations (for $k=2,3,4$ we get respectively sequences A64098 (which is also connected to Markov numbers, Frieze-patterns, cluster algebras and so on), A72878 and A72879 of the OEIS).

For $k=2$ (and $s_0=s_1=s_2=1$), the evaluations at $q=1$ and $q=2$ give A165903 and A72882 of the OEIS, other evaluations seem to be absent from the OEIS.

Answer 1

Recurrences like these often times have conserved quantities. For your particular case the quantity $$A_n=\frac{s_{n+1}+qs_n+qs_{n-k+1}+s_{n-k}}{s_ns_{n-1}\cdots s_{n-k+1}}$$ remains constant for any $n\geq k$.

Proof: We can calculate the following induction step $$s_{n+2}=\frac{s_{n+1}^2+s_n^2+\cdots +s_{n+2-k}^2+qs_{n+1}s_n+\cdots qs_{n+3-k}s_{n+2-k}}{s_{n+1-k}}$$ $$=\frac{s_{n+1}s_{n-k}+s_{n+1}^2-s_{n-k+1}^2+qs_{n+1}s_n-qs_{n-k+1}s_{n-k+2}}{s_{n+1-k}}$$ $$=\frac{s_{n+1}s_{n-k}+s_{n+1}(A_ns_n\cdots s_{n-k+1}-qs_n-qs_{n-k+1}-s_{n-k})-s_{n-k+1}^2+qs_{n+1}s_n-qs_{n-k+1}s_{n-k+2}}{s_{n+1-k}}$$ $$=A_ns_{n+1}\cdots s_{n-k+2}-qs_{n+1}-qs_{n-k+2}-s_{n-k+1}$$ which ultimately gives us $A_{n+1}=A_n$.

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This means that your sequence can also be generated with the following recurrence relation $$s_{n+1}==A_ks_{n}\cdots s_{n-k+1}-qs_{n}-qs_{n-k+1}-s_{n-k}$$ and since $A_k$ is a fixed Laurent monomial in the initial $s_i$'s, your claim follows.