Matthias Beck

activity

20012026

most citedMaximal Periods of (Ehrhart) Quasi-Polynomials

29 citations · 154 across the 72 of their papers we have counts for

collaborators

Showing math.NTShow all

21 papers · 1 filter

math.NT2019

Frobenius Coin-Exchange Generating Functions

Leonardo Bardomero, Matthias Beck

We study variants of the \emph{Frobenius coin-exchange problem}: given $n$ positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnega…

math.NT2018

Weighted lattice point sums in lattice polytopes, unifying Dehn--Sommerville and Ehrhart--Macdonald

Matthias Beck, Paul E. Gunnells, Evgeny Materov

Let $V$ be a real vector space of dimension $n$ and let $M \subset V$ be a lattice. Let $P \subset V$ be an $n$ -dimensional polytope with vertices in $M$ , and let $φ\colon V\rightarr…

math.NT2016

Reciprocity Theorems for Bettin--Conrey Sums

Juan S. Auli, Abdelmejid Bayad, Matthias Beck

Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k…

math.NT2014

Partitions with fixed differences between largest and smallest parts

George E. Andrews, Matthias Beck, Neville Robbins

We study the number $p (n, t)$ of partitions of $n$ with difference $t$ between largest and smallest parts. Our main result is an explicit formula for the generating function $P_t(q)…

math.NT2014

Variations on a Generating-Function Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence

Matthias Beck, Neville Robbins

A \Def{composition} of a positive integer $n$ is a $k$ -tuple $(ł_{1}, ł_{2}, \dots, ł_{k}) \in Z_{> 0}^{k}$ such that $n = ł_{1} + ł_{2} + \dots + ł_{k}$ . Our goal is to enumerate those composi…

math.NT2013

Rademacher-Carlitz Polynomials

Matthias Beck, Florian Kohl

We introduce and study the \emph{Rademacher-Carlitz polynomial} \[ \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k \] wh…

math.NT2013

Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

Abdelmejid Bayad, Matthias Beck

The \emph{Barnes $ζ$ -function} is \[ ζ_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $ℜ (x) > 0$ and $\Re(z) >…

math.NT2011

Mahonian Partition Identities Via Polyhedral Geometry

Matthias Beck, Benjamin Braun, Nguyen Le

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $Ω$ operator to systematically comp…

math.NT2010

Bernoulli--Dedekind Sums

Matthias Beck, Anastasia Chavez

Let $p_{1}, p_{2}, \dots, p_{n}, a_{1}, a_{2}, \dots, a_{n} \in N$ , $x_{1}, x_{2}, \dots, x_{n} \in R$ , and denote the $k$ th periodized Bernoulli polynomial by $\B_k(x)$. We study expressions of the form \…

math.NT2010

An Extreme Family of Generalized Frobenius Numbers

Matthias Beck, Curtis Kifer

We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_{0}$ that cannot be represented as a nonnegati…

math.NT2008

Finite Trigonometric Character Sums Via Discrete Fourier Analysis

Matthias Beck, Mary Halloran

We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan'…

math.NT2007

Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra

Matthias Beck, Christian Haase, Asia R. Matthews

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] v^(k-1), where u and v are indeterminates and a and b are positive int…

math.NT2005

Determinations of Analogues of Gauss Sums and Other Trigonometric Sums

Matthias Beck, Bruce C. Berndt, O-Yeat Chan +1

Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the…

math.NT2003

Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube

Matthias Beck, Sinai Robins, Shelemyahu Zacks

Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the f…

math.NT2003

The Reciprocity Law for Dedekind Sums via the constant Ehrhart coefficient

Matthias Beck

We obtain a new motivated proof of the reciprocity law for Dedekind sums by computing the constant coefficient of the Ehrhart polynomial for a rectangular triangle in two ways. On…

math.NT2003

Refined upper bounds for the linear Diophantine problem of Frobenius

Matthias Beck, Shelemyahu Zacks

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d…

math.NT2002★ 5 cited

An extension of the Frobenius coin-exchange problem

Matthias Beck, Sinai Robins

Given positive integers $a_{1}, ..., a_{n}$ with $g cd (a_{1}, ..., a_{n}) = 1$ , we call an integer t representable if there exist nonnegative integers $m_{1}, ..., m_{n}$ such that $t = m_1 a_1 + ..…

math.NT2002

Some experimental results on the Frobenius problem

Matthias Beck, David Einstein, Shelemyahu Zacks

We study the Frobenius problem: given relatively prime positive integers $a_{1}, ..., a_{d}$ , find the largest value of t (the Frobenius number) such that $\sum_{k = 1}^{d} m_{k} a_{k} = t$ has…

math.NT2002★ 1 cited

The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums

Matthias Beck, Ricardo Diaz, Sinai Robins

We study the number of lattice points in integer dilates of the rational polytope $P = (x_{1}, ..., x_{n}) \in R_{\geq 0}^{n} : \sum_{k = 1}^{n} x_{k} a_{k} \leq 1$ , where $a_{1}, ..., a_{n}$ are posit…

math.NT2001★ 3 cited

Dedekind sums: a combinatorial-geometric viewpoint

Matthias Beck, Sinai Robins

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattic…

math.NT2001★ 1 cited

Arithmetic partition sums and orbits of Z_n^k under the symmetric group S_k

Matthias Beck, Alex J. Feingold, Michael D. Weiner

We study M(n,k,r), the number of orbits of {(a_1,...,a_k)\in Z_n^k | a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arith…