Linear Recurrences and Power Series Division

Bousquet-Mélou and Petkovšek investigated the generating functions of multivariate linear recurrences with constant coefficients. We will give a reinterpretation of their theory by means of division theorems for formal power series, which clarifies the structural background and provides short, conceptual proofs. In addition, extending the division to the context of differential operators, the case of recurrences with polynomial coefficients can be treated in an analogous way. Throughout this paper we will use the following notation: Let K be a field and d be the number of variables. Bold letters indicate tuples x = (x1, . . . , xd), monomials are written as x = x1 1 . . . x nd d , and the scalar product is denoted by u ·w = u1v1 + · · · + udvd. The support supp(F (x)) of a formal power series F (x) = ∑ n∈N fnx n ∈ KJxK is the set of all monomials x whose coefficients fn are nonzero. Let KJxK 6≥p denote the set of all power series with support in N\(p+N). When we speak of a weight vector, we mean a vector in R with positive, Q-linearly independent components. A weight vector w induces a total order ≺w on Z d as well as on the monomials x in KJxK: a ≺w b and x a ≺w x b if w · a < w · b. The initial monomial inw (F ) of a power series F w.r.t. to a weight vector w is defined to be the ≺w -minimal element of supp(F ). 1. Recurrences with constant coefficients Let (fn )n∈Nd be a sequence in K given by the recurrence