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.

Abstract. For any positive integer n and variables a and x we define the generalized Legendre polynomial Pn(a, x) by Pn(a, x) = Pn k=0 a k −1−a k ( 1−x 2 ). Let p be an odd prime. In this paper we prove many congruences modulo p related to Pp−1(a, x). For example, we show that Pp−1(a, x) ≡ (−1)〈a〉p Pp−1(a,−x) (mod p), where a is a rational p− adic integer and 〈a〉p is the least nonnegative resid...

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.

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[f ] when f(x) has general algebraic-logarithmic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the forms f(x) ∼ ∞ ∑ ...

We analyze the problem facing the team that wins the toss at the deciding fifth set of a volleyball match. The team’s decision to serve or to receive the service can make a difference to the eventual outcome of the match. We characterize the conditions under which it is better to serve or to receive the service at that set. These conditions are obtained by first expressing the exact probability...

In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocatio...

The analyzing power of pp → ppπ 0 reaction has been measured at the beam energy of 390 MeV. The missing mass technique of final protons has been applied to identify the π 0 production event. The dependences of the analyzing power on the pion emission-angle and the relative momentum of the protons have been obtained. The angular dependence could be decomposed by the Legendre polynomial and the r...

Newman polynomials are those with all coefficients in {0, 1}. We consider here the problem of finding Newman polynomials P such that all the coefficients of P 2 are so small as possible for deg P and P (1) given. A set A ⊂ [1, N ] is called a B2[g] sequence if every integer n has at most g distinct representations as n = a1 + a2 with a1, a2 ∈ A and a1 ≤ a2. Gang Yu [4] introduced a new idea to ...

Brightness temperature difference (BTD) values are calculated for selected Geostationary Operational Environmental Satellite (GOES-6) channels (3.9, 12.7 pm) and Advanced Very High Resolution Radiometer channels (3.7, 12.0 pm). Daytime and nighttime discrimination of particle size information is possible given the infrared cloud extinction optical depth and the BTD value. BTD values are present...

Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. We analyze the coercivity on R of multivariate polynomials f ∈ R[x] in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on the vertex set of their Newton polytopes. These cond...