Guessing Linear Recurrence Relations of Sequence Tuplesand P-recursive Sequences with Linear Algebra

Abstract

Given several n-dimensional sequences, we first present an algorithm for computing the Grobner basis of their module of linear recurrence relations. A P-recursive sequence (ui)i ∈ Nn satisfies linear recurrence relations with polynomial coefficients in i, as defined by Stanley in 1980. Calling directly the aforementioned algorithm on the tuple of sequences ((ij, ui)i ∈ Nn)j for retrieving the relations yields redundant relations. Since the module of relations of a P-recursive sequence also has an extra structure of a 0-dimensional right ideal of an Ore algebra, we design a more efficient algorithm that takes advantage of this extra structure for computing the relations. Finally, we show how to incorporate Grobner bases computations in an Ore algebra K t1,...,tn,x1,...,xn, with commutators xk,xl-xl,xk=tk,tl-tl,tk= tk,xl-xl,tk=0 for k ≠ l and tk,xk-xk,tk=xk, into the algorithm designed for P-recursive sequences. This allows us to compute faster the elements of the Grobner basis of which are in the ideal spanned by the first relations, such as in 2D/3D-space walks examples.