# Generating functions and generalized Dedekind sums

@article{Gessel1997GeneratingFA, title={Generating functions and generalized Dedekind sums}, author={Ira M. Gessel}, journal={Electron. J. Comb.}, year={1997}, volume={4} }

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…

## 53 Citations

Dedekind sums: a combinatorial-geometric viewpoint

- MathematicsUnusual Applications of Number Theory
- 2000

This expository paper shows that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes, and obtains and generalizes reciprocity laws of Dedkind, Zagier, and Gessel.

Dedekind Sums, the Building Blocks of Lattice-Point Enumeration

- Mathematics
- 2015

We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1 They…

The Arithmetic of Rational Polytopes

- Mathematics
- 2000

We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of…

The Ring of Malcev-Neumann Series and the Residue Theorem

- Mathematics
- 2004

We develop a theory of the field of double Laurent series, iterated Laurent series, and Malcev-Neumann series that applies to most constant term evaluation problems. These include (i) MacMahon's…

Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind—Rademacher Sums

- MathematicsDiscret. Comput. Geom.
- 2002

This work gives explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon, and shows that Gessel's reciprocity law is a special case of the one for Dedekind—Rademacher sums, due to Rademacher.

The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

- Mathematics
- 2002

We study the number of lattice points in integer dilates of the rational polytope P={(x1,…,xn)∈R⩾0n:∑k=1nxkak⩽1}, where a1,…,an are positive integers. This polytope is closely related to the linear…

The Coin-Exchange Problem of Frobenius

- Mathematics
- 2015

Suppose we are interested in an infinite sequence of numbers \(\left (a_{k}\right )_{k=0}^{\infty }\) that arises naturally from geometric problems, or from recursively defined problems. Is there a…

A Gallery of Discrete Volumes

- Mathematics
- 2015

A unifying theme of this book is the study of the number of integer points in polytopes, where the polytopes live in a real Euclidean space \(\mathbb{R}^{d}\). The integer points \(\mathbb{Z}^{d}\)…

Counting Lattice Points in Polytopes: The Ehrhart Theory

- Mathematics
- 2015

Given the profusion of examples that gave rise to the polynomial behavior of the integer-point counting function \(L_{\mathcal{P}}(t)\) for special polytopes \(\mathcal{P}\), we now ask whether there…

## References

SHOWING 1-10 OF 21 REFERENCES

Higher dimensional dedekind sums

- Mathematics
- 1973

In this paper we will study the number-theoretical properties of the expression v1 nkal rcka,, d(p; a I . . . . . an) = ( 1) n/2 ~ cot cot (1) k=l P P and of related finite trigonometric sums. In Eq.…

Invariants of finite groups and their applications to combinatorics

- Mathematics
- 1979

1 CONTENTS 1. Introduction 2. Molien's theorem 3. Cohen-Macaulay rings 4. Groups generated by pseudo-reflections 5. Three applications 6. Syzygies 7. The canonical module 8. Gorenstein rings 9.…

Nörlund’s Number Bn (n)

- Mathematics
- 1993

The Bernoulli numbers of order k may be defined by [10, p. 143]
$$\left( {\frac{x} {{e^x - 1}}} \right)k = \sum\limits_{n = 0}^\infty {{\text{B}}_n^{{\text{(}}k{\text{)}}} } \frac{{x^n }} {{n!}}$$…

N orlund's number B(n) n , Applications of Fibonacci Numbers, Vol

- 5 (G. E. Bergum, A. N. Philippou, and A. F. Horadam, eds.), Kluwer Acad. Publ., Dordrecht,
- 1993

A sequence of polynomials related to roots of unity, Problem E 3339, Solution

- Math. Monthly
- 1991

A sequence of polynomials related to roots of unity, Problem E 3339, Solution by R

- Math. Monthly
- 1991

Asymptotic formulas from Chebyshev polynomials , Problem 6332

- Amer . Math . Monthly
- 1994

Asymptotic formulas from Chebyshev polynomials, Problem 6332, Solution by H

- J. Seiffert, Amer. Math. Monthly
- 1994

Asymptotic formulas from Chebyshev polynomials, Problem 6332, Solution by H

- J. Seiiert, Amer. Math. Monthly
- 1994