Related papers: Twisted Mahler discrete residues
We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some…
In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such…
A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that…
In difference algebra, summability arises as a basic problem upon which rests the effective solution of other more elaborate problems, such as creative telescoping problems and the computation of Galois groups of difference equations. In…
In this paper, we give a new proof of a result due to Bezivin that a D-finite Mahler function is necessarily rational. This also gives a new proof of the rational-transcendental dichotomy of Mahler functions due to Nishioka. Using our…
We give another proof of a result of Adamczewski and Bell concerning Mahler equations: A formal power series satisfying a $p-$ and a $q-$Mahler equation over ${\mathbb C}(x)$ with multiplicatively independent positive integers $p$ and $q$…
Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently,…
For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{\lambda_n\}_{n=1}^\infty$ satisfying $|\lambda_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals…
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at…
This is the second part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. From the lifting theorem obtained in the first part, we first derive a…
E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions…
We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this…
In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be…
We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to…
After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic…
We present the stellar resolution, a "flexible" tile system based on Robinson's first-order resolution. After establishing formal definitions and basic properties of the stellar resolution, we show its Turing-completeness and to illustrate…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove…
This paper is a plea for diagonals and telescopers of rational, or algebraic, functions using creative telescoping, in a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and…
Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a…