Asymptotic expansions for the recurrence xn+1=1n∑k=1nfxkk$$ {x}_{n+1}=\frac{1}{n}\sum \limits_{k=1}^nf\left(\frac{x_k}{k}\right) $$. (16th August 2022)

• Record Type: Journal Article
• Authors: Popa, Dumitru
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 2(2023)
• Page Start: 2165
• Page End: 2173
• Publication Date: 2022-08-16
• Subjects: asymptotic expansion of a function -- asymptotic expansion of a sequence -- Cesáro lemma -- recursive sequences -- Stolz–Cesáro lemma; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8634
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : Let a ∈ − ∞, 0 $$ a\in \left[-\infty, 0\right) $$, f : a, ∞ → 0, ∞ $$ f:\left(a, \infty \right)\to \left(0, \infty \right) $$ be a function and the sequence x n n ≥ 1 $$ {\left({x}_n\right)}_{n\ge 1} $$ defined by x 1 > a $$ {x}_1>a $$ and x n + 1 = 1 n ∑ k = 1 n f x k k $$ {x}_{n+1}=\frac{1}{n}\sum \limits_{k=1}^nf\left(\frac{x_k}{k}\right) $$ for every n ≥ 1 $$ n\ge 1 $$ . We prove that: If f $$ f $$ is decreasing and continuous at 0 then, lim n → ∞ x n = f 0 $$ \underset{n\to \infty }{\lim }{x}_n=f(0) $$ ; if f $$ f $$ is decreasing and differentiable at 0 then, lim n → ∞ n ln n x n − f 0 = f 0 f ′ 0 $$ \underset{n\to \infty }{\lim}\frac{n}{\ln n}\left({x}_n-f(0)\right)=f(0){f}^{\prime }(0) $$ ; if f $$ f $$ is decreasing and twice differentiable at 0 then, there exists A = lim n → ∞ n x n − f 0 − f 0 f ′ 0 ln n ∈ ℝ $$ A=\underset{n\to \infty }{\lim}\left[n\left({x}_n-f(0)\right)-f(0){f}^{\prime }(0)\ln n\right]\in \mathbb{R} $$ . Moreover, x n = f 0 + f 0 f ′ 0 ln n n + A n + f 0 f ′ 0 1 − f ′ 0 ln n n 2 + 2 A 1 − f ′ 0 − B 2 n 2 + o 1 n 2 $$ {\displaystyle \begin{array}{cc}\hfill {x}_n& =f(0)+\frac{f(0){f}^{\prime }(0)\ln n}{n}+\frac{A}{n}+\frac{f(0){f}^{\prime }(0)\left(1-{f}^{\prime }(0)\right)\ln n}{n^2}\hfill \\ {}\hfill & \kern10pt +\frac{2A\left(1-{f}^{\prime }(0)\right)-B}{2{n}^2}+o\left(\frac{1}{n^2}\right)\hfill \end{array}} $$ where B = 2 f 0 f ′ 0 2 + f 0 f ′ 0 + f 0 2 f ′ ′ 0 $$ B=2f(0){\left[{f}^{\prime }(0)\right]}^2+f(0){f}^{\prime }(0)+{\left[f(0)\right]}^2{f}^{\prime \prime }(0) $$ . Some concrete applications are given.