The usual method of separation of variables to find a basis of solutions of Laplace’s equation in toroidal coordinates is particularly appropriate for axially symmetric applications; for example, to find the potential outside a charged conducting torus. An alternative procedure is presented here that is more appropriate where the boundary conditions are independent of the spherical coordinate y...

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xxo, xx¡, ...} with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Lague...

In this paper, a Legendre sequence of period p for any odd prime p is explicitely represented as a sum of trace functions from G F(2n) to G F(2), where n is the order of 2 mod p.

We consider a fourth-order eigenvalue problem on a semi-infinite strip which arises in the study of viscoelastic shear flow. The eigenvalues and eigenfunctions are computed by a spectral method involving Laguerre functions and Legendre polynomials.

This note reports on the recent advancements in the search for explicit representation , in classical special functions, of the solutions of the fourth-order linear ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.

In the main theorem of this paper we show that any involution on the class of lower semi-continuous convex functions which is order-reversing, must be, up to linear terms, the well known Legendre transform.

We derive analogues of theorems of Ramanujan, Koshliakov and Guinand for primitive characters. As particular examples, transformation formulas involving the Legendre symbol and sums-of-divisors functions are established.

We prove a conjecture of Cuttler et al. (2011) on the monotonicity of normalized Schur functions under the usual (dominance) partialorder on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions. © 2015 Elsevier Ltd. All rights reserved. We prove a conjecture of Cuttler et al. [1] on themonotonicity of normalized Schur fun...

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 ...

In this report, we present two mathematical results which can be useful in a variety of settings. First, we present an analysis of Legendre polynomials triple product integral. Such integrals arise whenever two functions are multiplied, with both the operands and the result represented in the Legendre polynomial basis. We derive a recurrence relation to calculate these integrals analytically. W...