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.

in this study, a total of 101,147 monthly test day records of somatic cell count collected from 13,977 first lactation holstein cows (in 183 herds) calved between 2002 and 2006 were used. for the genetic analysis, a random regression test day model was utilized. somatic cell score (scs) was calculated based on natural logarithm of somatic cell count. in the model, fixed effect of contemporary g...

We show how Turán’s inequality Pn(x) −Pn−1(x)Pn+1(x) ≥ 0 for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality |x|Pn(x) 2 − Pn−1(x)Pn+1(x) ≥ 0, −1 ≤ x ≤ 1, effortlessly with the aid of our method.

Longevity, productive life, or lifespan of dairy cattle is an important trait for dairy farmers, and it is defined as the time from first calving to the last test date for milk production. Methods for genetic evaluations need to account for censored data; that is, records from cows that are still alive. The aim of this study was to investigate whether these methods also need to take account of ...