Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

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.

To appear in Communications in Contemporary Mathematics

References in corpus (3)

• Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
• Tannakian approach to linear differential algebraic groups
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

Cited by in corpus (5)

• Computing differential Galois groups of second-order linear $q$-difference equations
• Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
• Mahler discrete residues and summability for rational functions
• Computing discrete residues of rational functions
• Twisted Mahler discrete residues