Related papers: Classical multiple orthogonal polynomials for arbi…
The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and…
A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main…
A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we…
Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for…
We study a new family of q-Meixner multiple orthogonal polynomials of the first kind. The discrete orthogonality conditions are considered over a non-uniform lattice with respect to different q-analogues of Pascal distributions. We address…
We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the…
Classical orthogonal polynomials have widespread applications including in numerical integration, solving differential equations, and interpolation. Changing basis between classical orthogonal polynomials can affect the convergence,…
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…
We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{\mu-\frac12}$, $\mu > -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first…
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…
Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
For the oscillator-like systems, connected with the Laguerre, Legendre and Chebyshev polynomials coherent states of Glauber-Barut-Girardello type are defined. The suggested construction can be applied to each system of orthogonal…
We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic…
The multivariable version of ordinary and generalized Hermite polynomials are the natural solutions of the classical heat equation and of its higher order versions. We derive the associated Burgers equations and show that analogous…
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for…
This paper is devoted to characterizations classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal…
In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and…
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…