MULTIVARIATE NEWTON SUMS: IDENTITIES AND GENERATING FUNCTIONS

Trigonometric Identities and Sums of Separable Functions

Modern computers have made commonplace many calculations that were impossible to imagine a few years ago. Still, when you face a problem with a high physical dimension, you immediately encounter the Curse of Dimensionality [1, p.94]. This curse is that the amount of computing power that you need grows exponentially with the dimension. The simplest manifestation of this curse appears when you tr...

Generating functions and generalized Dedekind sums

We study sums of the form ∑ ζ R(ζ), where R is a rational function and the sum is over all nth roots of unity ζ (often with ζ = 1 excluded). We call these generalized Dedekind sums, since the most well-known sums of this form are Dedekind sums. We discuss three methods for evaluating such sums: The method of factorization applies if we have an explicit formula for ∏ ζ(1− xR(ζ)). Multisection ca...

Derived From Multivariate Generating Functions

Inequalities and Identities Involving Sums of Integer Functions

The floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. More precisely, the floor function ⌊x⌋ is the largest integer not greater than x and the ceiling function ⌈x⌉ is the smallest integer not less than x. Iverson [7, p. 127] introduced this notation and the terms floor and ceiling in the early 1960’s and now this notation is s...

Binomial Sums and Identities