On Positivity and Minimality for SecondOrder Holonomic Sequences
Abstract
An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as Precursive or Pfinite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle{v_n}\rangle_{n \in\mathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a secondorder recurrence $g_3(n)u_n = g_2(n)u_{n1} + g_1(n)u_{n2}$, where each coefficient $g_3, g_2,g_1 \in \mathbb{Q}[n]$ is a polynomial of degree at most $1$. We establish two main results. First, we show that deciding positivity for such sequences reduces to deciding minimality. And second, we prove that deciding minimality is equivalent to determining whether certain numerical expressions (known as periods, exponential periods, and periodlike integrals) are equal to zero. Periods and related expressions are classical objects of study in algebraic geometry and number theory, and several established conjectures (notably those of Kontsevich and Zagier) imply that they have a decidable equality problem, which in turn would entail decidability of Positivity and Minimality for a large class of secondorder holonomic sequences.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.12282
 Bibcode:
 2020arXiv200712282K
 Keywords:

 Mathematics  Number Theory;
 Computer Science  Discrete Mathematics;
 11B37;
 11Y65;
 68R99;
 G.2.1
 EPrint:
 38 pages