An infinite sequence
<un>n ∈ 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 un ≥ 0, and
minimal if, given any other linearly independent sequence
<vn>n ∈ N
satisfying the same recurrence relation,
the ratio un/vn
converges to 0.
In this paper, we focus on holonomic sequences
satisfying a second-order recurrence
g3(n)un = g2(n)un-1 + g1(n)un-2,
where each coefficient
g3, g2, g1 ∈ 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.
Submitted, 2020. 38 pages.
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© 2020 George Kenison, Oleksiy Klurman, Engel
Lefaucheux, Florian Luca, Pieter Moree,
JoëlOuaknine, Markus A. Whiteland, and James Worrell.