相关论文: On Bochner-Krall orthogonal polynomial systems
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth…
From our work on partition functions in log gravity, we show that the palindromic numerators in two variables of bigraded symmetric orbifold Hilbert series take the form of sums of products of Kostka-Foulkes polynomials associated with a…
We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…
We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions.…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial…
Matrix spherical functions associated to the compact symmetric pair $(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2)\times \mathrm{U}(m))$, having reduced root system of type $\mathrm{BC}_2$, are studied. We consider an irreducible…
The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring of translational operators in the spectral parameter. In the rational limit, these…
We wish to investigate the $D_{\omega}$-classical orthogonal polynomials, where $D_{\omega}$ is a special case of the Hahn operator. For this purpose, we consider the problem of finding all sequences of orthogonal polynomials such that…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important applications involving structured low rank matrices.…
A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the…
Rational function approximations find applications in many areas including macro-modeling of high-frequency circuits, model order reduction for controller design, interpolation and extrapolation of system responses, surrogate models for…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
We argue that operads provide a general framework for dealing with polynomials and combinatory completeness of combinatory algebras, including the classical $\mathbf{SK}$-algebras, linear $\mathbf{BCI}$-algebras, planar…
This paper intends to construct discrete spectral transformations for Cauchy-Jacobi orthogonal polynomials, and find its corresponding discrete integrable systems. It turns out that the normalization factor of Cauchy-Jacobi orthogonal…
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…