This paper is devoted to implementing the Legendre spectral collocation method to introduce numerical solutions of a certain class of fractional variational problems (FVPs). The properties of the Legendre polynomials and Rayleigh-Ritz method are used to reduce the FVPs to the solution of system of algebraic equations. Also, we study the convergence analysis. The obtained numerical results show ...

The main object of this paper is to give a di erent approach to proof of generating matrix functions for Laguerre matrix polynomials. We also obtain the hypergeometric matrix representations, addition theorem, nite summation formula and an integral representation for Laguerre matrix polynomials. We get the relations between Laguerre, Legendre and Hermite matrix polynomials. We get the generatin...

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

Let μ be a probability measure on the real line with finite moments of all orders. Suppose the linear span of polynomials is dense in L(μ). Then there exists a sequence {Pn}∞ n=0 of orthogonal polynomials with respect to μ such that Pn is a polynomial of degree n with leading coefficient 1 and the equality (x − αn)Pn(x) = Pn+1(x) + ωnPn−1(x) holds, where αn and ωn are SzegöJacobi parameters. In...

In this study test-day records of milk (kg), fat (g), and protein (g) yields, somatic cell score (SCS, cells/ML) collected by Animal Breeding Center of Iran during 2007 and 2009 were used to estimate genetic parameters using random regression model. Models with different order of Legendre polynomials were compared using Bayesian information criterion (BIC).For milk, fat yield and SCS genetic an...

In this paper I propose to estimate densities with possibly restricted support semi-nonparametrically via semi-nonparametric (SNP) estimation of densities on the unit interval. The latter will be done similarly to the approach of Gallant and Nychka (1987), but instead of using Hermite polynomials I propose to use orthonormal Legendre polynomials on the unit interval. This approach will be appli...

The classical orthogonal functions of mathematical physics are closely related to Lie groups. Specifically, they are matrix elements of, or basis vectors for, unitary irreducible representations of lowdimensional Lie groups. We illustrate this connection for: The Wigner functions, spherical harmonics, and Legendre polynomials; the Bessel functions; and the Hermite polynomials. These functions a...