Some applications of Legendre numbers
type: Item
instance of: scholarly article
title: Some applications of Legendre numbers
zbMATH Open document ID: 0641.33010
DOI: 10.1155/S0161171288000481
author: Paul W. Haggard
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46266
review text: For n and s nonnegative integers, the associated Legendre functions are defined by \[ P^ s_ n(x)=(1-x^ 2)^{s/2}(d^ s/dx^ s)P_ n(x), \] where \(P_ n(x)\) is the Legendre polynomial of degree n. The Legendre numbers are defined by \(P^ k_ n=P^ k_ n(0)\). From these the associated Legendre polynomials are obtained and the derivatives of these polynomials at \(x=0\) are derived using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials and \(x^ n\) in series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is given.
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 26C99
zbMATH DE Number: 4044220
zbMATH Keywords: associated Legendre functions
zbMATH Keywords: Legendre polynomial
zbMATH Keywords: Pascal numbers
reviewed by: Matthew F. Wyneken
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4044220