First-order factors of linear Mahler operators

TL;DR

This work develops two complete algorithms for factoring linear Mahler operators to obtain first-order right-hand factors, i.e., hypergeometric solutions, by relating factorization to Riccati Mahler equations. The first method extends Petkovšek’s Gosper–Petkovšek framework to Mahler operators; the second uses Hermite–Padé approximants to reconstruct ramified rational solutions from series data, producing parametrized candidate solutions that are then validated. A rigorous structural foundation is provided via a difference-ring framework, Puiseux and ramified-series solutions, and explicit degree- and ramification-bounds, enabling controlled searches and termination proofs. The authors implement and benchmark both approaches, showing that the HP-based reconstruction often outperforms the naive Petkovšek-style search, and apply the results to differential transcendence criteria for Mahler functions. The work thereby advances practical tools for analyzing Mahler equations, with implications for automatic sequences, transcendence theory, and differential-algebraic independence.

Abstract

We develop and compare two algorithms for computing first-order right-hand factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M + \ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$ for some integer $b \geq 2$. In other words, we give algorithms for finding all formal infinite product solutions of linear functional equations$\ell_r(x) f(x^{b^r}) + \dots + \ell_1(x) f(x^b) + \ell_0(x) f(x) = 0$. 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-Pad{é} 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.

Figures (1)

• Figure 1: The inclusion relations for a difference field extension $F$ of $\mathbb K(x)$ and a ring extension $D$ of it satisfying Hypothesis \ref{['def:propertyP']}. The two rings $F$ and $D$ share the same field of constants $\mathbb L$. All inclusion but the dashed ones are difference ring inclusions; the dashed ones are only inclusions of $\mathbb L$-vector spaces. The space $\ker_D(L)$ is the space of the solutions of the operator $L$ underlying the equation \ref{['eq:linear-sigma']}.

Theorems & Definitions (148)

• Example 1.1
• Definition 1.2
• Definition 1.4
• Example 1.5
• Lemma 2.1
• proof
• Definition 2.2
• Definition 2.3
• Lemma 2.5
• proof