In this paper we describe an efficient algorithm for computing the potentials of the form r−λ where λ ≥ 1. This treecode algorithm uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of Gegenbauer polynomials to represent r−λ in a manner analogous to the use of Legendre polynomials for the expansion of ...

The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count th...

The bulk-scattering properties of dust aerosols and clouds are computed for the community radiative transfer model (CRTM) that is a flagship effort of the Joint Center for Satellite Data Assimilation (JCSDA). The delta-fit method is employed to truncate the forward peaks of the scattering phase functions and to compute the Legendre expansion coefficients for re-constructing the truncated phase ...

INTRODUCTION Random regressions (RR) on Legendre polynomial (LP) of age have been used to model longitudinal data such as milk production and weights at different ages. Usually such models involve high degree polynomials, especially for direct permanent environmental effects and, consequently, a large number of parameters to be estimated. High order polynomials are associated with sampling prob...

An O(N(logN)2/ loglogN) algorithm for computing the discrete Legendre transform and its inverse is described. The algorithm combines a recently developed fast transform for converting between Legendre and Chebyshev coefficients with a Taylor series expansion for Chebyshev polynomials about equallyspaced points in the frequency domain. Both components are based on the FFT, and as an intermediate...

For the oscillator-like systems, connected with the Laguerre, Legendre and Chebyshev polynomials coherent states of Glauber-Barut-Girardello type are defined. The suggested construction can be applied to each system of orthogonal polynomials including classical ones as well as deformed ones.

In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...

Adrien-Marie Legendre (1752–1833) made great contributions to analysis, number theory, celestial mechanics, and practical science. His name is attached to the Legendre differential equation, Legendre polynomials, the Legendre transformation, the Legendre symbol in number theory, the Legendre conditions in calculus of variations, the Legendre relation for elliptic integrals, the Legendre duplica...

Abstract. We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead t...