相关论文: The search for differential equations for orthogon…
New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, it gives an explicit solution to the Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit representations,…
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer…
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…
In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse…
Studies in thermodynamics often require the reduction of some first or second order partial derivatives in terms of a smaller basic set. A simple algorithm to perform such a reduction is presented here, together with a review of earlier…
In this article explicit formulas for the recurrence equation p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x) and the derivative rules sigma(x) p'_n(x) = alpha_n p_{n+1}(x) + beta_n p_n(x) + gamma_n p_{n-1}(x) and sigma(x) p'_n(x) =…
We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form $y^m = \phi(x)$ in $\mathbb{R}^2$ where $m = 1, 2$ and $\phi$ is a polynomial of arbitrary degree $d$, in terms of univariate semiclassical OPs. We compute…
We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we…
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach is based on the well-known fact that…
Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite…
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We…
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…
Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…
It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved…
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 unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second order difference or differential operator. The most…