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 $\{ a_n\}_{ n \ge 1}$.
Categories: math.CO