The rational component of the solution of a first-order linear recurrence relation with a rational right side

Differential Galois theory of linear difference equations

“…Many of these results appear implicitly (and some explicitly) in the literature [1,6,26,29,34,37,42,43,44] but we assemble them here in a form that meets our needs. We begin by considering the field C(x), C algebraically closed, with the difference operator σ(x) = x + 1 or σ(x) = qx, q not a root of unity.…”

“…With this assumption, we have Proposition 2.4 (Propositions 6.14 and 6.16) Let k be a σ∂-field with k σ a differentially closed field. There exists a σ∂-PV ring for (1) and it is unique up to σ∂-k-isomorphism. Furthermore, R σ = k σ .…”

“…As in the usual theory of linear difference equations, once one has selected a fundamental solution matrix of (1) in R, this group may be identified with elements of GL n (k σ ). The next result states that both Aut σ∂ (R/k) and R have additional structure.…”

Simplifying Sums in Πς*-Extensions

“…Given (Q(k), σ) with σ(k) = k + 1 , we can use any of the algorithms from [Abr75,Pau95] to compute the Σ-pair (f , g) = (…”

“…For the rational case this refined telescoping approach has been considered in [Abr75]; here the minimality of f is defined by the degree of the denominator polynomial. Theoretical inside and additional algorithmic approaches have been derived in [Pau95].…”