On generalized Dedekind sums involving quasi-periodic Euler functions

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...

A hybrid mean value involving a new Gauss sums and Dedekind sums

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.

Asymptotic Formulas and Generalized Dedekind Sums

Twisted Dedekind Type Sums Associated with Barnes’ Type Multiple Frobenius-Euler l-Functions

The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes’ type multiple FrobeniusEuler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes’ type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define gene...

Dedekind Sums with Arguments Near Euler ’ s Number e

We study the asymptotic behaviour of the classical Dedekind sums s(m/n) for convergents m/n of e, e2, and (e+1)/(e−1), where e = 2.71828 . . . is Euler’s number. Our main tool is the Barkan-Hickerson-Knuth formula, which yields a precise description of what happens in all cases.