Twisted Mahler discrete residues

TL;DR

This paper introduces twisted Mahler discrete residues, extending previous work on Mahler residues, to determine when rational functions can be expressed in a form involving a twisted Mahler difference, with applications to telescoping and Galois theory.

Contribution

The paper develops $ ext{λ}$-twisted Mahler discrete residues, providing a complete obstruction criterion for twisted Mahler summability and exploring initial applications in telescoping and Galois theory.

Findings

Defined $ ext{λ}$-twisted Mahler residues and proved their completeness.

Connected twisted residues to differential creative telescoping.

Applied residues to the Galois theory of Mahler equations.

Abstract

Recently we constructed Mahler discrete residues for rational functions and showed 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 rational function $g(x)$ and an integer $p > 1$. Here we develop a notion of $\lambda$-twisted Mahler discrete residues for $\lambda\in\mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^\lambda g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations.