A non-commutative discrete first Painlevé hierarchy: the Lax pair approach

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.