Legendre polynomials, Legendre--Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression
type: Item
instance of: scholarly article
title: Legendre polynomials, Legendre--Stirling numbers, and the left-definite spectral analysis of the Legendre differential expression
zbMATH Open document ID: 1014.33003
DOI: 10.1016/S0377-0427(02)00582-4
author: Lance L. Littlejohn
author: W. N. Everitt
author: R. Wellman
published in: Journal of Computational and Applied Mathematics
publication date: 2003-02-23T00:00:00Z
review text: The authors apply left-definite theory to the self-adjoint Legendre differential operator \(A(K)\), generated by the classical second order Legendre differential equation \(\ell_{L,K}[y](t)= -(1-t^2)y''+ 2ty'+ky=\lambda y\), \(t\in (-1,1)\), \(K>0\) with eigenfunctions as Legendre polynomials. The right-definite setting in this case is the Hilbert space \(H=L^2(-1,1)\). They determine the corresponding unique left-definite self-adjoint operator \(A_n(K)\) and characterize its domain in terms of another left-definite space. The key to determining these spaces and inner products is in finding the explicit Lagrangian symmetric form of the integral composite powers of \(\ell_{L,K} [\cdot]\). The key to determining these powers is a new identity involving a double sequence of numbers which they call Legendre-Stirling numbers.
reviewed by: R. S. Dahiya
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1872882
zbMATH Keywords: spectral theorem
zbMATH Keywords: left-definite Sobolev space
zbMATH Keywords: left-definite self-adjoint operator
zbMATH Keywords: Lagrangian symmetric
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: Legendre-Stirling numbers
MaRDI profile type: Publication
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full work available at URL: s0377-0427(02)00582-4
OpenAlex ID: W2071793397
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description: scientific article; zbMATH DE number 1872882