Log-concavity of $P$-recursive sequences
We consider the higher order Turán inequality and higher order log-concavity for sequences $\{a_n\}_{n \ge 0}$ such that \[ \frac{a_{n-1}a_{n+1}}{a_n^2} = 1 + \sum_{i=1}^m \frac{r_i(\log n)}{n^{α_i}} + o\left( \frac{1}{n^β} \right), \] where $m$ is a nonnegative integer, $α_i$ are real numbers, $r_i(x)$ are rational functions of $x$ and \[ 0 < α_1 < α_2 < \cdots < α_m < β. \] We will give a sufficient condition on the higher order Turán 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 Turán inequality holds.
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