The constant in asymptotic expansions for a cubic recurrence

1.   Introduction

Asymptotic analysis has a wide range of applications in the complexity of algorithms ^[1], solutions of all kinds of equations (e.g., ^[2,3,4]), combinatorial mathematics ^[5], some economic, biological or physical models ^[6].

Let $I$ be an interval of $\mathbb{R}$ and ${\cal C}(I, I)$ consist of all continuous functions $f$: $I\rightarrow I$. The $n$-th iterate $f^n$ of $f \in {\cal C}(I, I)$ is defined by

and

for all $x \in I$ recursively. Some researchers have given nice asymptotic expansions of iterates of elementary functions (e.g., ^[7,8]), some special functions (e.g., ^[9,10]), and some recursive sequences (e.g., ^[11,12]). It is known from ^[13,14,15] that

Lemma 1. Let

for all $n\geq 1$, where $f$: $(0, 1)\to (0, 1)$,

$a > 0$, $p > 0$. Then

Furthermore, Stević ^[16] obtained the first two terms of the asymptotic expansion of $x_n$ for three cases ^[17]. In particular, Ionascu and Stanica ^[17] obtained the first six terms of the asymptotic expansion of

in which, there appears a constant $C$ depending on the initial value $x_1$. And they have obtained some properties of Bruijn ([18, Section 8.6], also ^[7]), who obtained the first six terms of the asymptotic expansion of

in which, there also appears a constant $C$ depending on the initial value $x_1$.

In many asymptotic expansions, a special constant $C$ often appears, which is related to the initial value and can be regard as a function of the initial value

say $C(x)$. Such constants do not have closed form expressions, but can be expressed in the limiting form of a sequence. Studying $C(x)$ gives an insight into the dependence of the limiting behavior on the initial value. However, it is difficult to study the properties of such $C(x)$ in asymptotic expansions. First, $C(x)$ is very sensitive to these iterated functions. Second, $C(x)$ has no analytic closed-form expression. As far as we know, there is no general method to study it.

In order to discover some properties of these constants in asymptotic expansions, we consider the iterates of the cubic, i.e.,

The aim of this paper is to find the limiting expression of such $C(x)$ in the asymptotic expansion, to prove the monotonicity, differentiability of $C(x)$ and the functional equation that is satisfied.

The remainder of this paper is structured as follows. Section 2 presents some preliminary results on asymptotic analysis. Section 3 gives the first six terms of the asymptotic expansion of iterates of the cubic function is given, where $C(x)$ does not appear in a closed form, but only in a limiting form. And we prove some properties of $C(x)$. A short conclusion is given in the last section.

2.   Preliminaries

To compute the asymptotic expansion, we need the asymptotic estimates of the following sums.

Lemma 2. For $n\to \infty$, we have

Proof. For every $n\geq1$, let define

We have

By the Stolz-Cesáro Lemma, we have

It follows that

This completes the proof. □

Lemma 3. For $n\to \infty$, we have

Proof. The result follows immediately from the Euler Maclaurin formula

This completes the proof. □

The following lemma gives the first two terms of the asymptotic expansion of $x_n$ for Eq (1.1).

Lemma 4. Let

for all $n\in \mathbb{N}$, where

Then the following results hold:

${\rm (i)}$ $\underset{n\to\infty}{\lim}\sqrt{n}x_{n} = \dfrac{1}{\sqrt{2}}$.

${\rm (ii)}$ $\underset{n\to\infty}{\lim}\dfrac{n^{3/2}}{\ln n}(x_{n}-\dfrac{1}{\sqrt{2n}}) = -\dfrac{3}{8\sqrt{2}}$.

Proof. (ⅰ) It directly follows from Lemma 1.

(ⅱ) Since $x_{n} > 0$, the equation

is rewritten into

Let $a_n = x_n^{-1}.$ Then

Putting

we have

Thus

Since

one can see that $b_{n}\to\infty$ for $n\to\infty$. It follows that

as $n\geq n_{0}$. So there exists a positive integer $m$ such that $b_{n}\geq mn$. Then

and

Substituting

into (2.1), we have

where $r_n$ satisfies

Consequently,

This completes the proof. □

The following result comes from [19, Corollary 2.2.3].

Lemma 5. Let $\left\{a_j\right\}$ be a sequence of non-negative real numbers. Then the series $\sum_{j = 1}^{\infty} a_j$ and the product $\prod_{j = 1}^{\infty}\left(1-a_j\right)$ either both converge or both diverge.

3.   Main results

First, we give the first six terms of the asymptotic expansion of $x_n$ for Eq (1.1).

Theorem 1. Let

for all integers $n\geq 1$ where

Then there exists a constant $C\in \mathbb{R}$ such that

where $C$ depends on the initial value $x_1 = x$. Moreover

where

Proof. We will continue to use those symbols and expressions in the proof of Lemma 4. Since the series $\sum_{k = 1}^{+\infty}\left(r_{k+1}-r_k\right)$ is convergent, say $C$. It follows from Lemma 2 that

According to (2.2), let

It follows from (2.1) that

Thus

It follows that

It follows from Lemmas 2 and 3, and

that

Then

Therefore,

where $C$ depends on the initial value, and

Let the initial value $x_1 = x$. From the asymptotic expansion of $x_n$, one can see that $C$: $(0, 1)\to \mathbb{R}$ is given by

□ Next, we will discuss the monotonicity and smoothness of $C(x)$.

Theorem 2. The function $C(x)$ is strictly decreasing on $(0, \frac{\sqrt{3}}{3}),$ strictly increasing on $(\frac{\sqrt{3}}{3}, 1)$, and its minimum value

Proof. By the proof of Lemma 4,

satisfies the recurrence relation

where

It follows that

And $h^n(x)$ is a strictly increasing sequence with respect to $n$. Furthermore, one can see that

for all $x\in(1/2, +\infty)$ and every $n\in \mathbb{N}$.

According to the proof of Theorem 1, let us define

Obviously,

Thus we can consider $g(x)$ instead of $C(x)$. One can obtain

since

and

Therefore $g^{\prime}(x) > 0$ for $x > \frac{3}{2}$ and $g^{\prime}(x) < 0$ for

It follows from

that $C(x)$ is strictly decreasing on $(0, \frac{\sqrt{3}}{3}),$ strictly increasing on $(\frac{\sqrt{3}}{3}, 1)$, and $C(x)$ takes the minimum value at ${\frac{\sqrt{3}}{3}}$. □

See Figures 1 and 2, which are plotted with Matlab. Figure 2 is a magnified view of the central section shown in Figure 1.

Theorem 3. Let

for $x\in \left(1/2, \infty\right)$ and

Then $g$ is continuously differentiable and

Proof. From the previous discussion, one can see that

According to the inequality

and Lemma 5, the sequence of $(h^{n-1})^{\prime}(x)$ is absolutely convergent. Therefore the sequence $h^{n-1} (1)+\int_{1}^{x}(h^{n-1})^{\prime}(t)dt$ converges to

This fact shows that $g$ is continuously differentiable.

The functional equation

immediately, follows from

This completes the proof. □

Corollary 1. The function $C(x)$ is continuously differentiable on the interval $\left(0, 1\right)$.

Proof. The result follows directly from the composition

This completes the proof. □

4.   Conclusions

Our analysis gives the first six terms of the asymptotic expansion of iterates of the cubic function

where the constant $C$ with respect to the initial value $x = x_1$

One can see that $C(x)$ does not have a closed-form expression, but only a limiting form. We study the properties of $C(x)$ through the iterated function. This is a natural and tractable approach. It is proved that $C(x)$ is continuously differentiable on the interval $(0, 1)$, and strictly decreasing on $(0, \frac{\sqrt{3}}{3})$, strictly increasing on $(\frac{\sqrt{3}}{3}, 1)$, with minimum value $C(\frac{\sqrt{3}}{3})$.

Our approach to studying $C(x)$ for the cubic function provides an example for studying and understanding some properties of constants in the other asymptotic expansions.

Author contributions

Xiaoyu Luo: writing original draft, methodology, proof of conclusions; Yong-Guo Shi: writing original draft, methodology, proof of conclusions; Kelin Li: validation, writing review, editing, proof of conclusions; Pingping Zhang: validation, writing review, editing, proof of conclusions. The authors contributed equally to this work. All the authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to thank the reviewers for having read this manuscript very carefully and for their many constructive and valuable comments, which have enhanced the final version of this paper.

Yong-Guo Shi is supported by NSF of Sichuan Province (2023NSFSC0065). Pingping Zhang is supported by NSF of Shandong Province (ZR2021MA044), Scientific Research Fund of Binzhou University (BZXYLG2118).

Conflict of interest

The authors declare no conflicts of interest.