Positivity of arbitrary-order P-recursive sequences with a unique dominant root

TL;DR

This paper provides a sufficient condition to determine the ultimate positivity of P-recursive sequences with a unique dominant root, including methods to verify positivity through initial terms, with practical examples.

Contribution

It introduces a new criterion for positivity of high-order P-recursive sequences and demonstrates its application with concrete examples.

Findings

Established a sufficient condition for ultimate positivity.

Showed how to verify positivity by checking initial terms.

Provided examples of higher-order P-recursive sequences.

Abstract

We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two.