The scaling properties and the multiple derivative of Legendre\n polynomials

2017/10/28 by Guillaume Laurent, Laurent, Guillaume Marc, Geoffrey Robert Harrison +1 · 1 citation
Chemistry · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1711.00925

openalex publication_date 2017/10/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the scaling properties of Legendre polynomials Pn(x).\nWe show that Pn(ax), where a is a constant, can be expanded as a sum of either\nLegendre polynomials Pn(x) or their multiple derivatives dkPn(x)/dxk, and we\nderive a general expression for the expansion coefficients. In addition, we\ndemonstrate that the multiple derivative dkPn(x)/dxk can also be expressed as a\nsum of Legendre polynomials and we obtain a recurrence relation for the\ncoefficients.\n

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