Inequalities Associated with the Root Sequences of $P$-Recursive Sequences

Abstract

The Turán inequalities and the Laguerre inequalities are closely related to the Laguerre-Pólya class and the Riemann hypothesis. These inequalities have been extensively studied in the literature. In this paper, we propose a method to determine a positive integer $N$ such that the sequences $\{\sqrt[n]{a_n}/n!\}_{n \ge N}$ and $\{\sqrt[n+1]{a_{n+1}}/(\sqrt[n]{a_n} n!)\}_{n \ge N}$ satisfy the higher order Turán inequality and the Laguerre inequality of order two for a $P$-recursive sequence $\{a_n\}_{n \ge 1}$.