Theoretical study demonstrates higher-order q-discrete Painlevé extensions via periodic cluster algebras, indicating integrability through co-primeness preservation.
We construct the q-discrete Painlevé I and II equations and their higher order analogues by virtue of periodic cluster algebras. Using particular k × k exchange matrices, we show that the cluster algebras corresponding to $k=4$ and $5$ give the q-discrete Painlevé I and II equations respectively. For k ≥ 6, we obtain higher-order discrete equations that satisfy an integrability criterion, namely, the co-primeness property.
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Naoto Okubo (2026) studied this question.
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