# On Inequality Decision Problems for Low-Order Holonomic Sequences

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

January, 2021

### Abstract

An infinite sequence $\langle u_n \rangle$ of real numbers is holonomic if it satisfies a linear recurrence relation with polynomial coefficients: $g_{k+1}(n)u_{n+k} = g_k(n)u_{n+k-1} + \cdots + g_1(n)u_n$ where each coefficient $g_0, \ldots, g_k \in \mathbb{Q}[n]$. Here $k$ is the order of the sequence; order-1 holonomic sequences are also known as hypergeometric sequences. The degree of the sequence is the highest degree of the polynomial coefficients appearing in the recurrence relation.
A holonomic sequence $\langle u_n \rangle$ is said to be 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 $0$. Given two hypergeometric sequences $\langle u_n \rangle$ and $\langle v_n \rangle$, the Hypergeometric Inequality Problem asks whether, for all $n$, $u_n \leq v_n$.
In this paper, we focus on various decision problems for second-order and hypergeometric sequences, and in particular on effective reductions concerning such problems. Some of these reductions also involve certain numerical quantities (known as periods, exponential periods, and pseudoperiods, originating from algebraic geometry and number theory), and classical decision problems regarding equalities among these quantities.
We establish the following:

- For second-order holonomic sequences, the Positivity Problem reduces to the Minimality Problem.
- For second-order, degree-1 holonomic sequences, the Positivity and Minimality Problems both reduce to the Equality Problems for exponential periods and pseudoperiods.
- The Hypergeometric Inequality Problem reduces to the Pseudoperiod Equality Problem.