Twisted Mahler discrete residues

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.

References in corpus (7)

• Hypertranscendence and linear difference equations
• Hypertranscendence of solutions of Mahler equations
• Computing solutions of linear Mahler equations
• Galois groups for integrable and projectively integrable linear difference equations
• Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
• Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
• Mahler discrete residues and summability for rational functions