Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
Carlos E. Arreche
TL;DR
The paper develops algorithms to compute the difference-differential Galois group $G$ for a second-order linear difference equation $\sigma^2(y)+a\sigma(y)+by=0$ with $a,b\in\bar{\mathbb{Q}}(x)$, by first obtaining the difference Galois group $H$ via Hendriks' method and then refining to $G\subseteq H$ using the difference-differential Galois theory of Hardouin–Singer. It provides a structured, case-based framework (diagonalizable, reducible, imprimitive, and containing $\mathrm{SL}_2$) to determine $G$, including explicit descriptions of unipotent radicals, determinants, and residue-based obstructions, and it describes how to translate $G$ into concrete differential-algebraic relations among the solutions. The work combines Riccati-analysis, discrete residues, and $\delta$-polynomial constraints to yield explicit relations and embedding data, culminating in representative examples that illustrate the computation of $G$ and the resulting differential relations. This yields a practical roadmap for extracting differential relations from the computed Galois group and demonstrates how the theory can handle a variety of structural scenarios, with potential applications to identifying differential dependencies among special function solutions of difference equations.
Abstract
We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation of the form $ y(x+2)+a(x)y(x+1)+b(x)y(x)=0,$ where the coefficients $a(x),b(x)\in \bar{\mathbb{Q}}(x)$ are rational functions in $x$ with coefficients in $\bar{\mathbb{Q}}$. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.