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...

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...

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...

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.

Let p be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for È p−1 2 k=0 2k k ¡ 2 m −k (mod p 2). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

Article history: Received 5 June 2009 Accepted 1 August 2009 Available online 7 August 2009

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) ∼ ∞ ∑ ...