Log-concavity of $P$-recursive sequences

Abstract

We consider the higher order Tur\'an inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{\alpha_i}} + o\left( \frac{1}{n^{\beta}} \right), \] where $m$ is a nonnegative integer, $\alpha_i$ are real numbers, $r_i(x)$ are rational functions of $x$ and \[ 0 < \alpha_1 < \alpha_2 < \cdots < \alpha_m < \beta. \] We will give a sufficient condition on the higher order Tur\'an inequality and the $r$-log-concavity for $n$ sufficiently large. Most $P$-recursive sequences fall in this frame. At last, we will give a method to find the exact $N$ such that for any $n>N$, the higher order Tur\'an inequality holds.