Related papers: On differential relations of 2-orthogonal polynomi…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…
The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving…
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
We study orthogonal polynomials for a weight function defined over a domain of revolution, where the domain is formed from rotating a two-dimensional region and goes beyond the quadratic domains. Explicit constructions of orthogonal bases…
In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two.
We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with…
Pseudo-differential operator equations with parameter are studied. Uniform separability properties and resolvent estimates are obtained in terms of fractional derivatives. Moreover, maximal regularity properties of the pseudo-differential…
We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal…
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…
We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the…
This paper studies properties of q-Jacobi polynomials and their duals by means of operators of the discrete series representations for the quantum algebra U_q(su_{1,1}). Spectrum and eigenfunctions of these operators are found explicitly.…
We introduce notions of a separated solution and of a simple symmetry that generates a differential operator on a smooth manifold. We prove that a differential operator on a two dimensional manifold has a separated solution if it has a…
A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal…
The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
In this paper, a link between $q$-difference equations, Jacobi operators and orthogonal polynomials is given. Replacing the variable $x$ by $ q^{-n}$ in a Sturm-Liouville $q$-difference equation we discovered the Jacobi operator. With…
Representations by linear integral operators on $L_p$ spaces over measure spaces are investigated for the polynomial covariance type commutation relations and more general two-sided generalizations of covariance commutation relations…
We develop general expressions for the raising and lowering operators that belong to the orthogonal polynomials of hypergeometric type with discrete and continuous variable. We construct the creation and annihilation operators that…
We derive orthogonality relations for discrete q-ultraspherical polynomials and their duals by means of operators of representations of the quantum algebra su_q(1,1). Spectra and eigenfunctions of these operators are found explicitly. These…