Related papers: Lax forms of the $q$-Painlev\'e equations
I present a $q$-analog of the discrete Painlev\'e I equation, and a special realization of it in terms of $q$-orthogonal polynomials.
It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations…
The paper shows the summability of formal solutions of some linear q-difference-differential equations by using q-Laplace and q-Borel summation method.
We build several matrix Lax pairs of ${\rm q-P_{\rm VI}}$ valid even when the two eigenvalues of the residue of the monodromy matrix at infinity are equal. Their elements are rational functions of the dependent variables.
In this paper, we study linear differential equations arising from $\lambda$- Changhee polynomials (or called degenerate Changhee polynomials) and give some explicit and new identities for the $\lambda$-Changhee polynomials associated with…
In this paper, we construct two lattices from the $\tau$ functions of $A_4^{(1)}$-surface $q$-Painlev\'e equations, on which quad-equations of ABS type appear. Moreover, using the reduced hypercube structure, we obtain the Lax pairs of the…
All Painlev\'e equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic).…
A polynomial deformation of the Kowalewski top is considered. This deformation includes as a degeneration a new integrable case for the Kirchhoff equations found recently by one of the authors. A $5\times 5$ matrix Lax pair for the deformed…
Recently, a birational representation of an extended affine Weyl group of type $A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}$ was proposed with the aid of a cluster mutation. In this article we formulate this representation in a…
We construct 2x2-matrix linear problems with a spectral parameter for the Painleve equations I-V by means of the degeneration processes from the elliptic linear problem for the Painleve VI equation. These processes supplement the known…
We consider some natural connections which arise between right-flat (p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form, with a "Lax p-tuple" of linear differential operators,…
Variants of the q-hypergeometric equation were introduced in our previous paper with Hatano. In this paper, we consider degenerations of the variant of the q-hypergeometric equation, which is a q-analogue of confluence of singularities in…
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…
We consider equations arising from rational Lax representations. A general method to construct recursion operators for such equations is given. Several examples are given, including a degenerate bi-Hamiltonian system with a recursion…
All possible 1-parametric classical and transcendent degenerated solutions of the fourth Painleve equation with the corresponding connection formulae of the asymptotic parameters are described.
Four 4-dimensional Painlev\'e-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlev\'e system.…
We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of…
The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of q-difference equations satisfied by the universal characters, which is an extension of the q-KP hierarchy…
We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x \in…
We will describe natural `Lax pairs' for the difference Painleve equations with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations.