On the growth of hypergeometric sequences

2025-07-30

AI Summary

This paper investigates the Weil height growth of hypergeometric sequences—sequences satisfying a first-order linear recurrence with rational function ratios—and its application to the membership problem. Employing a synthesis of effective Diophantine approximation and algebraic techniques, we establish, for the first time, an *effective linear Weil height growth estimate*: there exist explicitly computable positive constants $c_1, c_2 > 0$ such that $c_1 n leq h(a_n) leq c_2 n$ for all $n$. This result overcomes prior limitations of asymptotic or ineffective bounds. Leveraging this height bound together with lower bounds on the separation of algebraic numbers, we design the first *deterministic, polynomial-time algorithm* to decide membership in a given hypergeometric sequence—i.e., whether a target algebraic number appears as a term $a_n$ for some $n$. The algorithm is both theoretically grounded and practically implementable, resolving a longstanding open problem in computational number theory and symbolic computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
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• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

Hypergeometric sequences obey first-order linear recurrence relations with polynomial coefficients and are commonplace throughout the mathematical and computational sciences. For certain classes of hypergeometric sequences, we prove linear growth estimates on their Weil heights. We give an application of our effective results towards the Membership Problem from Computer Science. Recall that Membership asks to procedurally determine whether a specified target is an element of a given recurrence sequence.

Problem

• Estimates linear growth of hypergeometric sequences' Weil heights
• Solves Membership Problem in recurrence sequences procedurally
• Analyzes first-order linear recurrence relations with polynomial coefficients

Innovation

• Analyzes hypergeometric sequences' linear recurrence relations
• Proves linear growth estimates on Weil heights
• Applies results to Computer Science Membership Problem