Iterated integrals of Jacobi polynomials
type: Item
instance of: scholarly article
title: Iterated integrals of Jacobi polynomials
zbMATH Open document ID: 1443.33024
author: H. Pijeira Cabrera
author: Daniel Rivero-Castillo
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2020-04-23T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/33954
review text: The \(m\)th Jacobi polynomial with parameters \(\alpha,\beta\in\mathbb{R}\) is defined as \[P_{n}^{(\alpha, \beta)}(z)=\sum_{k=0}^{n}\binom{n+\alpha}{n-k}\binom{n+\beta}{k}\binom{2n+\alpha+\beta}{n}^{-1}(z-1)^{k}(z+1)^{n-k}.\] For a fixed \(m \in \mathbb{Z}_{+},\) let \(\mathcal{P}_{n, m}^{(\alpha, \beta)}\) be the monic polynomial of degree \(n+m\) given by \[\mathscr{P}_{n, m}^{(\alpha, \beta)}=P_{n+m}^{(\alpha-m, \beta-m)}.\] This polynomial is called the \(m\)th fundamental iterated integral of \(\frac{(n+m)!}{n!}P_n^{(\alpha, \beta)}\). There is renewed interest in the location of zeros of these polynomials, because this knowledge can be applied to the numerical solution of differential equations. Motivated by these applications, the authors give several theorems about the location if zeros of the \(\mathscr{P}_{n, m}^{(\alpha, \beta)}\) polynomials. Among many other nice results, they prove that the union of the zero sets of all these polynomials can be limited by a compact set of the complex plane.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 7192870
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: iterated integrals
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: asymptotic behavior
reviewed by: István Mező
MaRDI profile type: Publication
OpenAlex ID: W2972617958
cites work: Q4323301
cites work: Sobolev orthogonal polynomials: The discrete-continuous case
cites work: Sobolev–Gegenbauer-type orthogonality and a hydrodynamical interpretation
cites work: New shape functions for triangular \(p\)-FEM using integrated Jacobi polynomials
cites work: Sparse shape functions for tetrahedral \(p\)-FEM using integrated Jacobi polynomials
cites work: Orthogonality with respect to a Jacobi differential operator and applications
cites work: Zeros of Iterated Integrals of Polynomials
cites work: Strong asymptotics for Jacobi polynomials with varying weights
cites work: Quasi-orthogonality with applications to some families of classical orthogonal polynomials.
cites work: Q4095322
cites work: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters
cites work: Orthogonality of Jacobi polynomials with general parameters
cites work: Q5507817
cites work: Zeros of Jacobi polynomials with varying non-classical parameters
cites work: Riemann--Hilbert analysis for Jacobi polynomials orthogonal on a single contour
cites work: On the Zeros of Jacobi Polynomials P ( | alpha n , | beta n ) n (X)
cites work: On polar Legendre polynomials
cites work: Q4779788
cites work: Q3267566
cites work: Q4787437
cites work: Q5727762
Wikidata QID: Q127238639
DOI: 10.1007/S40840-019-00831-8
description: scientific article; zbMATH DE number 7192870