We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type A, we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided dif...

A divided difference expansion with remainder for a general divided difference is derived that contains Floater’s recent derivative expansion as a special case. key words divided difference, univariate, remainder, interpolation

Tools for computational differentiation transform a program that computes a numerical function F(x) into a related program that computes F ′(x) (the derivative of F). This paper describes how techniques similar to those used in computational-differentiation tools can be used to implement other program transformations—in particular, a variety of transformations for computational divided differen...

The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.

Under general conditions, the equation g(x, y) = 0 implicitly defines y locally as a function of x. In this short note we study the combinatorial structure underlying a recently discovered formula for the divided differences of y expressed in terms of bivariate divided differences of g, by analyzing the number of terms an in this formula. The main result describes six equivalent characterizatio...

We propose a q-analog of classical plethystic conjectures due to Foulkes. In our conjectures, a divided difference of plethysms of Hall-Littlewood polynomials Hn(x; q) replaces the analogous difference of plethysms of complete homogeneous symmetric functions hn(x) in Foulkes’ conjecture. At q = 0, we get back the original statement of Foulkes, and we show that our version holds at q = 1. We dis...

This is a study of Favard interpolation—in which the nth derivatives of the interpolant are bounded above by a constant times the nth divided differences of the data—in the case the data is given on some subset of a rectangular lattice in R k. In some instances, depending on the geometry of this subset, we construct a Favard interpolant, and in other instances, we prove that none exists.

Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend these results to vector coefficient sums.

In this paper we derive a formula for divided differences of composite functions of several variables with respect to rectangular grids of points. Letting the points coalesce yields a chain rule for partial derivatives of multivariate functions. Math Subject Classification: 41A05, 65D05, 26A06

Abstract In the expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, the author looks back and analyses some inequalities, the complete monotonicity of several functions involving ratios of two gamma or q-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio...