ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
Vol. LXXIII, 1 (2004)
p. 75 – 87
Effective Asymptotics for Some
Nonlinear Recurrences and Almost Doubly-Exponential Sequences
E. Ionascu and P. Stanica
Abstract.
We develop a technique to compute asymptotic expansions for
recurrent sequences of the form $a_{n+1}=f(a_n)$, where
$f(x)=x-ax^{\alpha}+bx^{\beta} +o(x^{\beta})$ as $x\rightarrow 0$,
for some real numbers $\alpha, \beta$, $a$, and $b$ satisfying
$a>0$, $1<\alpha<\beta $. We prove a result which summarizes the
present stage of our investigation, generalizing the expansions in
[Amer. Math Monthly, Problem E $3034[1984,58]$, Solution
$[1986,739]$]. One can apply our technique, for instance, to
obtain the formula: $\displaystyle a_{n}={\sqrt{3}\over
\sqrt{n}}- {3\sqrt{3}\over10}{\ln n \over
n\sqrt{n}}+{9\sqrt{3}\over 50}{\ln n\over n^2\sqrt{n}}
+o\left({\ln n\over n^{5/2}}\right)$, where $a_{n+1}=\sin(a_{n})$,
$a_1\in \RR$. Moreover, we consider the recurrences
$a_{n+1}=a_n^2+g_n$, and we prove that under some technical
assumptions, $a_n$ is almost doubly-exponential, namely
$a_n=\lfloor{k^{2^n}}\rfloor$, $a_n=\lfloor{k^{2^n}}\rfloor+1$,
$a_n=\lfloor{k^{2^n}-\frac{1}{2}}\rfloor$, or
$a_n=\lfloor{k^{2^n}+\frac{5}{2}}\rfloor$ for some real number
$k$, generalizing a result of Aho and Sloane [Fibonacci Quart. 11
(1973), 429--437].
AMS Subject classification: 11B37, 11B83, 11K31, 11Y55, 34E05, 35C20, 40A05.
Keywords:
Sequences, Dynamics, Asymptotic Expansions, Doubly-Exponential.
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