In 1951, F. Brafman derived several “unusual” generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials Pn(x). His result was a consequence of Bailey’s identity for a special case of Appell’s hypergeometric function of the fourth type. In this paper, we present a generalization of Bailey’s identity and its implication to generating functions of Legendre po...

Hammerstein-Wiener systems present a structure consisting of three serial cascade blocks. Two are static nonlinearities, which can be described with nonlinear functions. The third block represents linear dynamic component placed between the first two Some common model structures include rational-type transfer function, orthogonal rational functions (ORF), finite impulse response (FIR), autoregr...

In this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. Almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. But we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as Legendre ...

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...

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...

A new family of self-calibration additional parameters (APs) is presented for calibrating airborne camera systems. We point out that photogrammetric self-calibration can to a very large extent be considered as a function approximation problem in mathematics. Based on the mathematical approximation theory, a novel family of so-called Legendre APs is developed based on the orthogonal Legendre pol...

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...

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...

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein–Bézier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The o...