was surmised already by Legendre and Gauss. However, it took a hundred years before the first proofs appeared, one by Hadamard and one by de la Vall~e Poussin (1896). Their and all but one of the subsequent proofs make heavy use of the Riemann zeta function. (The one exception is the long so-called elementary proof by Selberg [11] and Erd6s [41.) For Re s > 1 the zeta function is given by the D...

The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function Kν(x) = ∫ ∞ 0 e cosh t cosh νt dt when x > 0 and ν is complex, ν = α+ iβ. It is shown that Gaussian quadrature with weight function w(t) = exp(−et) on [0,∞] is a viable approach, unless x is small and/or β large, but in combination with Gauss–Legendre quadrature, even in these latter cases.

Gauss’s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form E(λ) : y = x(x− 1)(x− λ). We study a modular function which “measures” the variation of periods for the isomorphic curves E(λ) and E ( λ λ−1 ) , and we show that it padically “interpolates” the cusp form for the “congruent number” curve E(2), the case where these pairs...

The main aim of this article is to generalize the Legendre operational matrix to the fractional derivatives and implemented it to solve the nonlinear multi-order fractional differential equations. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used. The main characteristic behind the approach using this technique is that...

In this work, we will compare two approximation method based on hybrid Legendre and Block-Pulse functions and a computational method for solving nonlinear FredholmVolterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion.

The modified two-stream approximation is derived from the radiative transfer equation using a two-point gaussian quadrature and neglecting terms of order 2 > 3 in the Legendre expansion of the phase function. The results are compared with the Eddington approximation and with exact results for the special cases of perfect absorption, perfect scattering, and a semi-infinite layer with isotropic s...

We prove a Ramanujan-type formula for 520/π conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for 1/π.

A total of 20,065 weights recorded on 3016 Nelore animals were used to estimate covariance functions for growth from birth to 630 days of age, assuming a parametric correlation structure to model within-animal correlations. The model of analysis included fixed effects of contemporary groups and age of dam as quadratic covariable. Mean trends were taken into account by a cubic regression on orth...

Résumé Dans cet article nous allons présenter une méthode de détection d’objets basée sur l’analyse multifractale. La détection d’objets est particulièrement difficile lorsque l’on ne connaît pas les caractéristiques des objets pouvant être présents dans l’image. Contrairement aux méthodes le plus souvent proposées, cette méthode est destinée à détecter des objets artificiels dans des images ph...