Related papers: Discrete orthogonal polynomials and difference equ…

We are studying here the classical operator creating secondary polynomials associated with an orthogonal system for a continuous probability density function on a real interval. We know it is possible with the coupling of Stietjes…

The aim of these notes is to describe some recent results concerning dispersive estimates for principally normal pseudodifferential operators. The main motivation for this comes from unique continuation problems. Such estimates can be used…

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators…

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we…

Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…

In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained…

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calder\'on et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric…

The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a…

We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…

We look for spectral type differential equations for the generalized Jacobi polynomials and for the Sobolev-Laguerre polynomials. We use a method involving computeralgebra packages like Maple and Mathematica and we will give some…

The purpose of this note is to show how some results from the theory of partial differential equations apply to the study of pseudo-spectra of non-self-adjoint operators, which is a topic of current interest in applied mathematics.

Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the…

We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

The symmetric Al-Salam--Chihara polynomials for $q>1$ are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on $\ell^2(\Z)$ to which these polynomials are eigenfunctions. We determine…

A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…

We consider fractional differentiation operators in various senses and show that the strictly accretive property is the common property of fractional differentiation operators. Also we prove that the sectorial property holds for…

The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…

In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.

For a family of polynomials in two continuous variables, orthogonal with respect to a weight function, we prove, under suitable conditions, the equivalence of the following properties: the matrix Pearson equation of the weight, the second…