We analyzed 75,555 test-day milk flow records from 2175 primiparous Holstein cows that calved between 1997 and 2005. Milk flow was obtained by dividing the mean milk yield (kg) of the 3 daily milking by the total milking time (min) and was expressed as kg/min. Milk flow was grouped into 43 weekly classes. The analyses were performed using a single-trait Random Regression Models that included di...

Random regression models (RRM) have become common for the analysis of longitudinal data or repeated records on individual over time. The goal of this paper was to explore the use of random regression models with orthogonal / Legendre polynomials (RRL) to analyze new repeated measures called clutch size (CS) as a meristic trait for Iranian native fowl. Legendre polynomial functions of increasing...

Graphical inspection of multimodality is demonstrated using unsupervised lateral-inhibition neural networks. Three projection pursuit indices are compared on low dimensional simulated and real-world data: principal components 22], Legendre polynomial 6] and projection pursuit network 16].

We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman’s Legendre-Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1 − x) ∑∞ n=0 { n+k n } x analogous to that of the Eulerian numbers, where { n k } are Everitt, Littlejohn, and Wellman’s Legendre-Stirling numbers of the second kind. Finally we use a r...

This article is an attempt to understand the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from ...

The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count th...

We show that with small modifications in diverse expressions for the shifted Legendre polynomials, we can obtain the corresponding relations for Daubechies polynomials.

We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also correspo...

In this paper, summation formulae for the 2-variable Legendre polynomials in terms of certain multi-variable special polynomials are derived. Several summation formulae for the classical Legendre polynomials are also obtained as applications. Further, Hermite-Legendre polynomials are introduced and summation formulae for these polynomials are also established.