Positivity problem of three-term recurrence sequences

TL;DR

This paper investigates conditions under which three-term recurrence sequences are positive, providing criteria and applications such as positivity of Taylor coefficients and log-convexity, advancing understanding in sequence analysis.

Contribution

It introduces new necessary and sufficient conditions for the positivity problem and applies them to rational functions and sequence properties.

Findings

Established criteria for positivity and log-convexity of three-term recurrence sequences.

Proved positivity of diagonal Taylor coefficients for certain rational functions.

Provided a unified approach to analyze positivity in recurrence sequences.

Abstract

We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences.