  An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ 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 $u_n \geq 0$, 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 second-order recurrence $g_3(n)u_n
= g_2(n)u_{n-1} + g_1(n)u_{n-2}$, 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 period-like 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 second-order holonomic sequences.
