Related papers: Integrable Differential Systems for Deformed Lague…
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an…
We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three…
We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of…
We derive the integrability conditions of nonautonomous nonlinear Schr$\rm\ddot o$dinger equations using the Lax Pair and Similarity Transformation methods. We present a comparative analysis of these integrability conditions with those of…
We study reductions of the Korteweg--de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of $n$ second-order equations is obtained, which reduces to the Painlev\'e…
We show how to deform separable Henon-Heiles system with isospectral Lax representation, related with the stationary flow of the $5th$-order KdV, to respective non-autonomous systems of Painleve type with isomonodromic Lax representation.
The generating function of Stieltjes-Carlitz polynomials is a solution of Heun's differential equation and using this relation Carlitz was the first to get exact closed forms for some Heun functions. Similarly the associated…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,\gamma)=e^{-z^2+tz}|z-t|^{\gamma}(A+B\theta(z-t))$, where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and $z\in\textbf{R}$. By using the…
We study the Hankel determinant and orthogonal polynomials with respect to the two-parameter weight function $$ w(x)=w(x;t_1, t_2):=\exp(-x^6-t_2 x^4-t_1 x^2),\qquad x\in\mathbb{R}, $$ with $t_1,\; t_2 \in \mathbb{R}$. This problem arises…
In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain an asymptotic…
Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi--orthogonality conditions. We obtain several characterizations for these…
The generalized Stieltjes--Wigert polynomials depending on parameters 0\le p<1 and 0<q<1 are discussed. By removing the mass at zero of the N-extremal solution concentrated in the zeros of the D-function from the Nevanlinna parametrization,…
In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the…
In \cite{hab-2016,hab-2017}, Habibullin \emph{et.al} proposed an approach to construct Lax pairs of a nonlinear integrable partial differential equation (PDE), where one is the linearized equation of the studied PDE and the other is the…
Laguerre polynomials are orthogonal polynomials defined on positive half line with respect to weight $e^{-x}$. They have wide applications in scientific and engineering computations. However, the exponential growth of Laguerre polynomials…
A generalisation of the Stieltjes relations for the Painlev\'e-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these…
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…
In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have…
We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation…