Related papers: The kernel polynomial method based on Jacobi polyn…
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…
Asymptotic approximations of Jacobi polynomials are given for large values of the $\beta$-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the…
We work out the expression of the generalized Bessel function of type B in the two-rank case. This is done using Dijskma and Koornwinder's product formula for Jacobi polynomials and the obtained expression is given by multiple integrals…
The analytic solutions of the one-dimensional Schroedinger equation for the trigonometric Rosen-Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention…
In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative…
We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence…
We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we…
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…
In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold…
The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\alpha,\beta,c))_n$ have been classically defined for $\alpha,\beta\in(-1,\infty)$, $c\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard.…
We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…
We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a…
Hahn polynomials of several variables can be defined by using the Jacobi polynomials on the simplex as a generating function. Starting from this connection, a number of properties for these two families of orthogonal polynomials are…
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…
For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i) $$ $(a, b > -1, \beta > 0) $ a…
Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that…
We study kernel quadrature rules with convex weights. Our approach combines the spectral properties of the kernel with recombination results about point measures. This results in effective algorithms that construct convex quadrature rules…