# On Positivity and Minimality for Second-Order Holonomic Sequences

G. Kenison,
O. Klurman,
E. Lefaucheux,
F. Luca,
P. Moree,
J. Ouaknine,
M. A. Whiteland,
J. Worrell

August, 2021

### Abstract

An infinite sequence $\langle u_n \rangle$ of real numbers is holonomic if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle v_n \rangle$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$ as $n$ tends to infinity. In this paper we establish 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.

Publication

*Mathematical Foundations in Computer Science, MFCS `21*