On Positivity and Minimality for Second-Order Holonomic Sequences

Abstract

An infinite sequence unn of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each un ≥ 0, and minimal if, given any other linearly independent sequence vnn satisfying the same recurrence relation, the ratio un/vn → 0 as n → ∞. In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.