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arXiv · 2308.16765

Twisted Mahler discrete residues

Carlos E. Arreche·Yi Zhang
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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 $λ$-twisted Mahler discrete residues for $λ\in\mathbb{Z}$, 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 $p^λg(x^p)-g(x)$ 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.

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Carlos E. Arreche, Yi Zhang. 2023-08-31. Twisted Mahler discrete residues. https://doi.org/10.1093/imrn%2Frnae238

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