On the growth of hypergeometric sequences

TL;DR

This paper investigates the growth behavior of hypergeometric sequences, providing new bounds on their Weil heights and applying these results to solve the Membership Problem in computational contexts.

Contribution

It offers the first effective linear growth estimates for hypergeometric sequences' Weil heights and applies these findings to computational membership testing.

Findings

Established linear growth bounds on Weil heights of certain hypergeometric sequences

Applied growth estimates to improve algorithms for the Membership Problem

Demonstrated the relevance of hypergeometric sequence analysis in computational complexity

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.