The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

TL;DR

This paper investigates the decidability of the Threshold Problem for hypergeometric sequences with quadratic polynomial parameters, establishing new results under algebraic and conjectural assumptions, and extending previous work on related membership problems.

Contribution

It provides decidability results for the Threshold Problem for hypergeometric sequences with quadratic polynomial coefficients, including conditional results based on Schanuel's conjecture, and extends prior findings on membership problems.

Findings

Decidability established for sequences with roots in quadratic extensions

Conditional decidability results assuming Schanuel's conjecture

Extension of previous results on membership problems for hypergeometric sequences

Abstract

Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, $\langle u_n \rangle_{n=0}^\infty$ is hypergeometric if it satisfies a first-order linear recurrence of the form $p(n)u_{n+1} = q(n)u_{n}$ with polynomial coefficients $p,q\in\mathbb{Z}[x]$ and $u_0\in\mathbb{Q}$. In this paper, we consider the Threshold Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n\rangle_{n=0}^\infty$ and a threshold $t\in\mathbb{Q}$, determine whether $u_n \ge t$ for each $n\in\mathbb{N}_0$. We establish decidability for the Threshold Problem under the assumption that the coefficients $p$ and $q$ are monic polynomials whose roots lie in an imaginary quadratic extension of $\mathbb{Q}$. We also establish conditional decidability results; for example, under the assumption that the coefficients $p$ and $q$ are monic polynomials whose roots lie in any number of quadratic extensions of $\mathbb{Q}$, the Threshold Problem is decidable subject to the truth of Schanuel's conjecture. Finally, we show how our approach both recovers and extends some of the recent decidability results on the Membership Problem for hypergeometric sequences with quadratic parameters.