New Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sums

The Elliptic Apostol-dedekind Sums Generate Odd Dedekind Symbols with Laurent Polynomial Reciprocity Laws

Abstract. Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent poly...

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...

Generalized Reciprocity Laws for Sums of Harmonic Numbers

We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra system Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations.

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...

Asymptotic Formulas and Generalized Dedekind Sums