Asymptotic r-log-convexity and P-recursive sequences. (July 2019)

• Record Type: Journal Article
• Authors: Hou, Qing-Hu; Zhang, Zuo-Ru
• Is Part Of: Journal of symbolic computation. Volume 93(2019)
• Page Start: 21
• Page End: 33
• Publication Date: 2019-07
• Subjects: r-Log-convexity -- P-recursive sequence -- Asymptotic expansion; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2018.04.012
• Languages: English
• ISSNs: 0747-7171

Abstract

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 .