Asymptotic r-log-convexity and P-recursive sequences. (July 2019) - Hou, Qing-Hu; Zhang, Zuo-Ru
Abstract: Let L be the operator given by L a n n ≥ 0 = a n + 1 2 − a n a n + 2 n ≥ 0 . A sequence a n n ≥ 0 is called asymptotically r -log-convex if L k a n n ≥ N are non-negative sequences for 1 ≤ k ≤ r and a certain integer N . Based on asymptotic analysis, we present a method for proving the asymptotic r -log-convexity of a sequence. As an application, we consider the problem of proving the asymptotic r -log-convexity of P-recursive sequences. We find that most P-recursive sequences are asymptotically r -log-convex for any non-negative integer r once they are asymptotically log-convex. Moreover, we show how to find an explicit integer N such that they are r -log-convex for n ≥ N .