Generating functions and generalized Dedekind sums.

Ira Gessel

The electronic journal of combinatorics, The Wilf Festschrift volume, Tome 4 (1997) no. 2

Résumé

We study sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $\zeta$ (often with $\zeta =1$ 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 $\prod_\zeta (1-xR(\zeta))$. Multisection can be used to evaluate some simple, but important sums. Finally, the method of partial fractions reduces the evaluation of arbitrary generalized Dedekind sums to those of a very simple form.

Détail

DOI : 10.37236/1326

Classification : 11F20, 05A15
Mots-clés : generalized Dedekind sums, factorization, multisection, partial fractions