First-order factors of linear Mahler operators
arxiv(2024)
摘要
We develop and compare two algorithms for computing first-order right-hand
factors in the ring of linear Mahler operatorsℓ_r M^r + … + ℓ_1 M +
ℓ_0where ℓ_0, …, ℓ_r are polynomials in x and Mx = x^b M
for some integer b ≥ 2. In other words, we give algorithms for finding all
formal infinite product solutions of linear functional equationsℓ_r(x)
f(x^b^r) + … + ℓ_1(x) f(x^b) + ℓ_0(x) f(x) = 0. The first of our
algorithms is adapted from Petkovšek's classical algorithm forthe 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 transcendence of power series solutions of
Mahler equations.
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