Related papers: Orthogonal polynomials: from Heun equations to Pai…
This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved…
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
In this paper, we show that the classical discrete orthogonal univariate polynomials (namely, Hahn polynomials on an equidistant lattice with unit weights) of sufficiently high degrees have extremely small values near the endpoints (we call…
A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is…
Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…
We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev…
We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic…
We consider polynomials orthogonal on $[0,\infty)$ with respect to Laguerre-type weights $w(x)=x^\alpha e^{-Q(x)}$, where $\alpha>-1$ and where $Q$ denotes a polynomial with positive leading coefficient. The main purpose of this paper is to…
The Ruijsenaars-Schneider systems are `discrete' version of the Calogero-Moser (C-M) systems in the sense that the momentum operator p appears in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation parameter)…
A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ]$…
The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the…
This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial…
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*}…
We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
We introduce a new factorized and resummed waveform for circularized, nonspinning, compact binaries that leverages on the solution of the Teukolsky equation once mapped into a confluent Heun equation. The structure of the solution allows…
General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular…
We study the linear problems in $z,t$ (time) associated to the Painlev\'e III$_3$ and III$_1$ equations when the Painlev\'e solution $q(t)$ approaches a pole or a zero. In this limit the problem in $z$ for the Painlev\'e III$_3$ reduces to…
We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional…
In this paper, we establishes a connection between noncommutative Laurent biorthogonal polynomials (bi-OPs) and matrix discrete Painlev\'e (dP) equations. We first apply nonisospectral deformations to noncommutative Laurent bi-OPs to obtain…