ARTICLE | doi:10.20944/preprints202302.0222.v2
Subject: Computer Science And Mathematics, Computational Mathematics Keywords: Gauss quadrature; integral approximation; continuous measure; discrete measure; mixed measure; orthogonal polynomials; recursion relation
Online: 17 February 2023 (15:11:43 CET)
Gauss quadrature integral approximation is extended to include integrals with a measure consisting of a continuous as well as a discrete component. That is, we give an approximation for the integral of a function plus its sum over a discrete weighted set.
COMMUNICATION | doi:10.20944/preprints201901.0284.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: tridiagonal representation; orthogonal polynomials; potential functions; asymptotics; recursion relation; spectrum formula
Online: 29 January 2019 (04:37:49 CET)
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.