Log-behavior of the root sequences of P-recursive sequences

Abstract

In recent years, Sun has proposed numerous conjectures regarding the log-concavity of root sequences $\{\sqrt[n]{a_n}}_{n\geqslant 1}$. We establish criteria for the asymptotic log-concavity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ and the asymptotic ratio log-convexity of $\{\sqrt[n]{a_n}}_{n\geqslant 1}$ for $P$-recursive sequences $\{\sqrt[n]{a_n}}_{n\geqslant{0}}$. Additionally, by the aid of symbolic computation, we present a systematic approach to determine the explicit integer $N$ such that the sequence $\{\sqrt[n]{a_n}}_{n\geqslant{N}}$ is log-concave and the sequence $\{\sqrt[n]{a_n}}_{n\geqslant N}$ is ratio log-convex.