The Threshold Problem for Hypergeometric Sequences with Quadratic Parameters

TL;DR

This work studies the Threshold Problem for hypergeometric sequences defined by a recurrence $p(n)u_{n+1}=q(n)u_n$ with polynomial coefficients. It proves decidability when $p$ and $q$ are monic and split over an imaginary quadratic extension of $\mathbb{Q}$, and establishes conditional decidability (under Schanuel's conjecture) when every irreducible factor of $pq$ is linear or quadratic. It also recovers and extends recent decidability results on the Membership Problem for hypergeometric sequences with quadratic parameters. The results bridge algebraic and transcendental techniques, highlighting connections to gamma-function relations and asymptotic analysis, and have potential impact on automated verification and formal languages by enabling effective threshold checks in these 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.