Eulerian polynomials and B-splines

Abstract

Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, Bsplines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.

Exploring Matrix Generation Strategies in Isogeometric Analysis

“…The relation between Eulerian numbers and B-splines seems to be a deep one, see [9,20] for more information. By symmetry I …”

“…We shall replace the quadrature by an approximation, by means of interpolation or quasi-interpolation, of the part of the integrands (8), (9) or (10) that contains the geometry mapping G and its derivatives. This strategy aims at reducing the number of evaluations needed in quadrature-based approaches as well as avoiding the need to solve non-linear systems in order to derive quadrature rules.…”

Umbral calculus associated with Frobenius-type Eulerian polynomials

“…Since the seminal work of Katriel [17], the combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators a † and a of a single-mode boson having the usual commutation relations [a, a † ] = aa † − a † a = 1, [a, a] = 0 and [a † , a † ] = 0 have been studied intensively since the seventies, see [14,15,17,18,[26][27][28][29][32][33][34], and references therein. From a more mathematical point of view, the consequences of the noncommutative calculus of operators has been considered, in particular, by Maslov [29].…”

“…Note that umbral calculus has an application in physics of gases (see [35]) and in group theory and quantum mechanics (see [21,22]). Umbral calculus, in particular, the Sheffer sequences, has also been applied to the normal ordering of expressions involving bosonic creation and annihilation operators ( [14,15]). …”

“…Throughout the paper, we assume that λ ∈ C with λ = 1. The Frobenius-type Eulerian polynomials of order r are also given by [14,15,[19][20][21][22]), (1.10) where r is a positive integer. In the particular case,…”