Related papers: Exceptional Hahn and Jacobi polynomials with an ar…
Exceptional points are singularities in the spectrum of non-Hermitian systems in which several eigenvectors are linearly dependent and their eigenvalues are equal to each other. Usually it is assumed that the order of the exceptional point…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal…
Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we…
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…
We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
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…
We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family…
This paper delves into classical multiple orthogonal polynomials with an arbitrary number of weights, including Jacobi-Pi\~neiro, Laguerre of both first and second kinds, as well as multiple orthogonal Hermite polynomials. Novel explicit…
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…
In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are not necessarily $>-1$. We establish orthogonality on a generic closed contour on a…
In this paper, we consider the second-order differential expression \ell [y](x)=(1-x^2)(-(y'(x))'+k(1-x^2)^(-1)y(x))(x\in(-1,1)). This is the Jacobi differential expression with non-classical parameters {\alpha} = {\beta}= -1 in contrast to…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete measures by suitable polynomials. This problem was…
The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials…
We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the…
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The…
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of…