Inequalities associated with the root sequences of P-recursive sequences

TL;DR

This paper introduces a method to identify when certain normalized root sequences of P-recursive sequences satisfy higher order Turán and Laguerre inequalities, which are linked to important conjectures in analysis and number theory.

Contribution

It provides a novel approach to determine the threshold index N for which these inequalities hold for P-recursive sequences.

Findings

Method to find N for inequalities satisfaction

Sequences satisfy higher order Turán inequalities beyond N

Sequences satisfy Laguerre inequalities of order two beyond N

Abstract

The Tur{\'a}n inequalities and the Laguerre inequalities are closely related to the Laguerre-P\'{o}lya 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 Tur{\'a}n inequalities and the Laguerre inequalities of order two for a P-recursive sequence $\{a_n\}_{n \ge 1}$.