We show how lattice paths and the re ection principle can be used to give easy proofs of unimodality results. In particular, we give a \one-line" combinatorial proof of the unimodality of the binomial coe cients. Other examples include products of binomial coe cients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.

‎This paper provides the fractional derivatives of‎ ‎the Caputo type for the sinc functions‎. ‎It allows to use efficient‎ ‎numerical method for solving fractional differential equations‎. ‎At‎ ‎first‎, ‎some properties of the sinc functions and Legendre‎ ‎polynomials required for our subsequent development are given‎. ‎Then‎ ‎we use the Legendre polynomials to approximate the fractional‎ ‎deri...

We give a general approach to N -periodic sequences over a finite field Fq constructed via a subgroup D of the group of invertible elements modulo N . Well known examples are Legendre sequences or the two-prime generator. For some generalizations of sequences considered in the literature and for some new examples of sequence constructions we determine the linear complexity.

Horadam [7], in a recent article, defined two sequences of polynomials Jn(x) and j„(x), the Jacobsthal and Jacobsthal-Lucas polynomials, respectively, and studied their properties. In the same article, he also defined and studied the properties of the rising and descending polynomials i^(x), rn(x), Dn(x)y and dn(x), which are fashioned in a manner similar to those for Chebyshev, Fermat, and oth...

We combine the Lie algebraic methods and the technicalities associated with the monomialty principle to obtain new results concerning Legendre polynomial expansions. c © 2007 Elsevier Ltd. All rights reserved.

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...

The Müntz–Legendre polynomials arise by orthogonalizing the Müntz system {xλ0 , xλ1 , . . . } with respect to the Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulas 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...

have many applications in pure mathematics as in other branches of science. They are named by Karlin and Szegő in [8], Turán-type inequalities because the first of these type of inequalities was introduced by Turán in [18]. More precisely, he used some results of Szegő in [17] to prove the previous inequality for x ∈ (−1, 1), where fn is the Legendre polynomial of degree n. This classical resul...

The use of polynomial functions to describe the average growth trajectory and covariance functions of Nellore and MA (21/32 Charolais+11/32 Nellore) young bulls in performance tests was studied. The average growth trajectories and additive genetic and permanent environmental covariance functions were fit with Legendre (linear through quintic) and quadratic B-spline (with two to four intervals) ...

properties of the hybrid of block-pulse functions and lagrange polynomials based on the legendre-gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known legendre interpolation operator. the uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. the applicabili...