Abstract:Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlevé hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlevé hierarchy.
From: Irina Bobrova [view email]
[v1]
Thu, 28 May 2026 10:17:54 UTC (47 KB)
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