Related papers: Orthogonal Polynomials from Hermitian Matrices
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal…
We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to…
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…
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 consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}\}_{k=0}^{n-1}$ that can be treated as a discrete…
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…
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…
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…
We study orthogonal polynomials on a fully symmetric planar domain $\Omega$ that is generated by a certain triangle in the first quadrant. For a family of weight functions on $\Omega$, we show that orthogonal polynomials that are even in…
An open problem about two new families of orthogonal polynomials was posed by Alhaidari. Here we will identify one of them as Wilson polynomials. The other family seems to be new but we show that they are discrete orthogonal polynomials on…
We calculate the eigenvalues of some two-dimensional non-Hermitian Hamiltonians by means of a pseudospectral method and straightforward diagonalization of the Hamiltonian matrix in a suitable basis set. Both sets of results agree remarkably…
We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical…
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…
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…
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…
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…