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- W4293064046 abstract "Let ◂+▸a∈[−∞,0)$$ ain left[-infty, 0right) $$, f:◂→▸(a,∞)→(0,∞)$$ f:left(a,infty right)to left(0,infty right) $$ be a function and the sequence ◂◽.▸(xn)n≥1$$ {left({x}_nright)}_{nge 1} $$ defined by x1>a$$ {x}_1>a $$ and ◂=▸◂◽.▸xn+1=1n◂∑▸∑k=1nf(xkk)$$ {x}_{n+1}=frac{1}{n}sum limits_{k=1}^nfleft(frac{x_k}{k}right) $$ for every n≥1$$ nge 1 $$. We prove that: If f$$ f $$ is decreasing and continuous at 0 then, ◂=▸limn→∞xn=f(0)$$ underset{nto infty }{lim }{x}_n=f(0) $$; if f$$ f $$ is decreasing and differentiable at 0 then, ◂lim▸limn→∞◂⋅▸nlnn◂()▸(xn−f(0))=◂⋅▸f(0)f′(0)$$ underset{nto infty }{lim}frac{n}{ln n}left({x}_n-f(0)right)=f(0){f}^{prime }(0) $$; if f$$ f $$ is decreasing and twice differentiable at 0 then, there exists A=◂+▸limn→∞[n◂()▸(xn−f(0))−◂⋅▸f(0)f′(0)lnn]∈ℝ$$ A=underset{nto infty }{lim}left[nleft({x}_n-f(0)right)-f(0){f}^{prime }(0)ln nright]in mathbb{R} $$. Moreover, xn=◂+▸f(0)+◂⋅▸f(0)f′(0)lnnn+An+◂⋅▸f(0)f′(0)◂()▸(1−f′(0))lnnn2◂+▸+◂/▸◂−▸2A◂()▸(1−f′(0))−B2n2+o(1n2)$$ {displaystyle begin{array}{cc}hfill {x}_n& =f(0)+frac{f(0){f}^{prime }(0)ln n}{n}+frac{A}{n}+frac{f(0){f}^{prime }(0)left(1-{f}^{prime }(0)right)ln n}{n^2}hfill {}hfill & kern10pt +frac{2Aleft(1-{f}^{prime }(0)right)-B}{2{n}^2}+oleft(frac{1}{n^2}right)hfill end{array}} $$ where ◂=▸B=◂⋅▸2f(0)◂[]▸[f′(0)]2+◂⋅▸f(0)f′(0)+◂⋅▸◂◽˙▸[f(0)]2f′′(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." @default.
- W4293064046 created "2022-08-26" @default.
- W4293064046 creator A5032304230 @default.
- W4293064046 date "2022-08-16" @default.
- W4293064046 modified "2023-09-27" @default.
- W4293064046 title "Asymptotic expansions for the recurrence xn+1=1n∑k=1nfxkk$$ {x}_{n+1}=frac{1}{n}sum limits_{k=1}^nfleft(frac{x_k}{k}right) $$" @default.
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- W4293064046 doi "https://doi.org/10.1002/mma.8634" @default.
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