论文标题
$ p $ recursursive序列的log-concavity
Log-concavity of $P$-recursive sequences
论文作者
论文摘要
我们考虑序列$ \ {a_n \} _ {n \ ge 0} $的较高阶段Turán不平等和更高级的log-concavity，以便\ [\ frac {\ frac {a_ {n-1} a_ {n-1} a_ {n + 1}}}}}} n）} {n^{α_i}} + o \ left（\ frac {1} {n^β} \ right），\]，$ m $是一个非负整数，$α_i$是实数，$ r_i（x）$是$ x $ and x $ x $ and cd [0 <α__1<α<α< β。 \]我们将在高级Turán不平等和$ r $ log-concavity上提供足够的条件，$ n $足够大。 大多数$ p $ - 示波序列都在此框架中。最后，我们将提供一种方法来找到确切的$ n $，以便对于任何$ n> n $，高级订单Turán不平等所拥有的。
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.
