Related papers: Exactly solvable periodic Darboux q-chains
The dressing chain is derived by applying Darboux transformations to the spectral problem of the Korteweg-de Vries (KdV) equation. It is also an auto-B\"acklund transformation for the modified KdV equation. We show that by applying Darboux…
We give a q-analogue of Gauss' divisibility theorem
I present a $q$-analog of the discrete Painlev\'e I equation, and a special realization of it in terms of $q$-orthogonal polynomials.
In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers…
We derive a $q$-analogue of the matrix sixth Painlev\'e system via a connection-preserving deformation of a certain Fuchsian linear $q$-difference system. In specifying the linear $q$-difference system, we utilize the correspondence between…
A general scheme is proposed for introduction of lattice and q-difference variables to integrable hierarchies in frame of $\bar{\partial}$-dressing method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov systems of…
A few years ago, the concept of a D-analogue was introduced as a Dirichlet series analogue for the already known and well researched hypergeometric q-series. The D-analogue of the q-Dixon sum is given here, in the context of seeing a direct…
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.
We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of…
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see…
The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].
A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If…
We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions, giving determinantal representations indexed by specific…
Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schr\"odinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In…
The dressing chain equations for factorizing operators of a spectral problem are derived. The chain equations itselves yield nonlinear systems which closure generates solutions of the equations as well as of the nonlinear system if both…
We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the…
In this paper a review is given of a class of sub-models of both approaches, characterized by the fact that they can be solved exactly, highlighting in the process a number of generic results related to both the nature of pair-correlated…
One of Darboux's seminal results is archived here
The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential…
In this work, the q-analogue of Bernoulli inequality is proved. Some other related results are presented.