Sep 18 – 29, 2023
Institut Henri Poincaré
Europe/Paris timezone

First-order factors of linear Mahler operators by Frédéric Chyzak

Sep 25, 2023, 3:00 PM
Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Amphithéâtre Hermite / Darboux

Institut Henri Poincaré

11 rue Pierre et Marie Curie 75005 Paris


Abstract. We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators ℓrMr+⋯+ℓ1M+ℓ0 where ℓ0,…,ℓr are polynomials in x and Mx=xbM for some integer b≥2. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations ℓr(x)f(xbr)+⋯+ℓ1(x)f(xb)+ℓ0(x)f(x)=0.
The first of our algorithms is adapted from Petkovšek's classical algorithm for the analogous problem in the case of linear recurrences. The second one proceeds by computing a basis of generalized power series solutions of the functional equation and by using Hermite-Padé approximants to detect those linear combinations of the solutions that correspond to first-order factors.
We present implementations of both algorithms and discuss their use in combination with criteria from the literature to prove the differential transcendance of power series solutions of Mahler equations.

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