The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

1.   Introduction

In the past years, many mathematicians were interested in finding the formula for the integer part of the reciprocal tails of the convergent series. That is, find the explicit value of $\left\lfloor\left(\sum_{k = n}^\infty a_k\right)^{-1}\right\rfloor$ when $\sum_{k = 1}^\infty a_k$ converges. The motivation of such research comes from the reciprocal sum of Fibonacci numbers. Let us recall that the Fibonacci sequence $\left(F_{n}\right)_{n\geq0}$ is defined by

where $n\geq0$. In ^[1], Ohtsuka and Nakamura proved

and

where $\left\lfloor x\right\rfloor$ denotes the greatest integer $\le x$. In ^[2], Wang proved

where $F_{-1} = F_1 = 1$. In ^[3], Hwang et al. provided the relevant formula for the reciprocal sum of the fourth power of the Fibonacci numbers, that is,

where $\{x\} = x-\lfloor x \rfloor$. In addition, some mathematicians studied the reciprocal sums of Fibonacci, Lucas, and Pell sequences, such as ^[4,5,6,7], while some mathematicians studied the reciprocal sums of other types of sequences, such as ^[8,9,10].

Many mathematicians also considered the asymptotic behavior of these sequences. That is, find a suitable function $g_n$ such that

when $\sum_{k = 1}^\infty a_k$ converges. Here the notation $A_n\sim B_n$ means that

In ^[11], Lee et al. proved that

for any positive integer $m$ and $0\le l\le m-1$. In ^[12], Marques et al. proved that for any positive integer $m$ there exists a positive constant $C_m$ such that

Moreover, they gave an explicit form for $C_m$ as follows:

where $L_n$ is the $n$th Lucas number. In ^[3], Hwang et al. studied the asymptotic behavior of the reciprocal sum of the fourth power of the Fibonacci numbers, and they proved that

In ^[13], Lee and Park studied the asymptotic behavior of the reciprocal sum of $F_kF_{k+m}$, and they proved that

and

where $l$ is a positive integer. In addition, some mathematicians studied other types of asymptotic behavior, such as ^[14,15]. Inspired by the above results, we use the method of Yuan et al. ^[16] to study the asymptotic behavior of the sequences that are more general than the Fibonacci sequence. Let $\left(u_n\right)_{n\geq0}$ be the special Lucas $u$-sequence defined by

where $n\geq0$, $B = \pm1$, and $A$ is an integer such that $A^2-4B > 0$. The relevant properties of the Lucas $u$-sequence can be found in Sun's book ^[17]. We know that the Binet formula is related to the sequence $\left(u_n\right)_{n\geq0}$ has the form

where

Let

where $m$ and $l$ are positive integers, $s = 1, 2, 3, 4$, and $C$ is any constant. The aim of this paper is to find a form $g_n$ such that

The rest of this paper is organized as follows: in Section 2, we give our main results. In Section 3, we give the proof of our main results.

2.   Main results

Let $u_n$ be defined by the second-order linear recurrence sequence (1.1). In this paper, we shall prove the following eight theorems.

Theorem 2.1. For any positive integer $m$, we have

Theorem 2.2. For any positive integer $m$, we have

where $C_{m} = \frac{2(1-B^m)}{(\alpha-\beta)^2}-\frac{2(\alpha^{2m}-1)^2}{{\left(\alpha-\beta\right)^2}\left(\alpha^{4m}-B^m\right)}$.

Theorem 2.3. For any positive integer $m$, we have

where $Q_m = \frac{u_m^2}{(1-(B\alpha)^{5m})(1-(B\beta)^{5m})}$.

Theorem 2.4. For any positive integer $m$, we have

where $U_m = \frac{u_m^2}{(1-B^m\alpha^{6m})(1-B^m\beta^{6m})}$ and $V_m = \frac{(\alpha^{4m}-1)^2}{(\alpha-\beta)^4} \left(\frac{16(\alpha^{4m}-1)}{(\alpha^{6m}-B^m)^2} -\frac{10}{\alpha^{8m}-1}\right)$.

Theorem 2.5. For all positive integers $m$ and $l$, we have

Theorem 2.6. For all positive integers $m$ and $l$, we have

Remark 2.1. Note that when $l = 1$, the two main terms of Theorems 2.5 and 2.6 are different. However, there is no contradiction since they are equivalent.

Theorem 2.7. For all positive integers $m$ and $l$, we have

(ⅰ)

where $C_{m, l} = u_l^2+\frac{2B^l}{A^2-4B}\left(1- \frac{(1-B^m)(\alpha^{4m}-B^m)}{(\alpha^{2m}-1)^2}\right)$.

(ⅱ)

where $C_{m, l}' = u_lu_{l-1}+ \frac{B^{l-1}}{A^2-4B}\left(A -\frac{2(\alpha-\alpha\beta^{2m})(\alpha^{4m}-B^m)} {(\alpha^{2m}-1)^2}\right)$.

Theorem 2.8. For any positive integer $m$ and constant $C$, we have

Let $A = 1$ and $B = -1$ in (1.1). Then $u_n$ becomes the $n$th Fibonacci number. So we can obtain some known results when we take some special values for $m, \, A, \, B$.

If we take $A = 1$ and $B = -1$ in Theorem 2.2, then

If $m$ is even, then it follows from (2.1) that

where $L_n$ is the $n$th Lucas number with $L_0 = 2, \, L_1 = 1$. If $m$ is odd, then it follows from (2.1) that

So we have the following corollary, which is given by ^[12].

Corollary 2.1. Let $F_n$ be the $n$th Fibonacci number with $F_0 = 0, \, F_1 = 1$ and let $L_n$ be the $n$th Lucas number with $L_0 = 2, \, L_1 = 1$. Then for any positive integer $m$, we have

where

If we take $m = A = 1$ and $B = -1$ in Theorem 2.4, then

and

which imply that

So we have the following corollary, which is given by [3, Corollary 4.3].

Corollary 2.2. Let $F_n$ be the $n$th Fibonacci number with $F_0 = 0, \, F_1 = 1$. Then we have

If we take $m = A = 1$ and $B = -1$ in Theorem 2.7, then

and

Note that

Then it follows from (2.2)–(2.4) that

and

So we have the following corollary, which is given by [13, Sections 3 and 4].

Corollary 2.3. Let $F_n$ be the $n$th Fibonacci number with $F_0 = 0, \, F_1 = 1$. Then for any positive integer $l$, we have

and

3.   Proof of the theorem

In this section, we give the proofs of the above eight theorems. We first give some identities that will be used in the proofs of our main results. Let $m$ and $k$ be positive integers. Then it follows from (1.2) that

Moreover, we have the following identities:

To prove the above eight theorems, we may split the proofs into eight subsections as follows:

3.1.   The proof of Theorem 2.1

In this subsection, we will provide a proof of Theorem 2.1.

Proof. By (3.1) and (3.2b), we have

Let $n$ be a positive integer. Then it follows from (3.3) that

Here the notation $f(x) = O(g(x))$ means that there is a constant $C$ such that $|f(x)|\leq Cg(x)$ for all large enough real numbers $x$. By (3.2a) and (3.4), we have

Then

So we have

□

3.2.   The proof of Theorem 2.2

In this subsection, we will provide a proof of Theorem 2.2.

Proof. By (3.1) and (3.2c), we have

where

Let $n$ be a positive integer. Then it follows from (3.5) that

where

Note that

Then we have

From (3.2a), (3.6), and (3.7), it follows that

where

Then

So we have

where $C_{m} = \frac{2(1-B^m)}{(\alpha-\beta)^2}-\frac{2(\alpha^{2m}-1)^2}{{\left(\alpha-\beta\right)^2}\left(\alpha^{4m}-B^m\right)}.$ □

3.3.   The proof of Theorem 2.3

In this subsection, we will provide a proof of Theorem 2.3.

Proof. By (3.1) and (3.2d), we have

where

Let $n$ be a positive integer. Then it follows from (3.8) that

where

Note that

Then we have

From (3.2a), (3.9), and (3.10), it follows that

where

Let $Q_m = \frac{u_m^2}{(1-(B\alpha)^{5m})(1-(B\beta)^{5m})}.$ Then we can obtain

Then

So we have

where $Q_m = \frac{u_m^2}{(1-(B\alpha)^{5m})(1-(B\beta)^{5m})}$. □

3.4.   The proof of Theorem 2.4

In this subsection, we will provide a proof of Theorem 2.4.

Proof. By (3.1) and (3.2e), we have

where

Let $n$ be a positive integer. Then it follows from (3.11) that

where

Note that

Then we have

By (3.2a), (3.12), and (3.13), we have

where

and

Let $U_m = \frac{u_m^2}{(1-B^m\alpha^{6m})(1-B^m\beta^{6m})}$. Then we can obtain

Then

So we have

where $U_m = \frac{u_m^2}{(1-B^m\alpha^{6m})(1-B^m\beta^{6m})}$ and $V_m = \frac{(\alpha^{4m}-1)^2}{(\alpha-\beta)^4} \left(\frac{16(\alpha^{4m}-1)}{(\alpha^{6m}-B^m)^2} -\frac{10}{\alpha^{8m}-1}\right)$. □

3.5.   The proof of Theorem 2.5

In this subsection, we will provide a proof of Theorem 2.5.

Proof. By (1.2) and (3.2b), we have

where

Let $n$ be a positive integer. Then it follows from (3.14) that

where

Then we have

By (3.2a), (3.15), and (3.16), we have

Then

So we have

□

3.6.   The proof of Theorem 2.6

In this subsection, we will provide a proof of Theorem 2.6.

Proof. By (1.2), we have

From (3.2b) and (3.17), it follows that

where

Let $n$ be a positive integer. Then it follows from (3.18) that

where

Then we have

By (3.2a), (3.19), and (3.20), we have

Then

So we have

□

3.7.   The proof of Theorem 2.7

In this subsection, we will provide a proof of Theorem 2.7.

Proof. Let $h$ be a positive integer. By (1.2), we have

where

Then we have

By (3.2b), (3.21), and (3.22), we have

Let $n$ be a positive integer. Then it follows from (3.23) that

where

Then we have

By (3.2a), (3.24), and (3.25), we have

(ⅰ) If we take $h = 2l$, then it follows from (3.26) that

where

Let $C_{m, l} = u_l^2+\frac{2B^l}{(\alpha-\beta)^2}\left(1- \frac{(1-B^m)(\alpha^{4m}-B^m)}{(\alpha^{2m}-1)^2}\right)$. Then

So we have

where $C_{m, l} = u_l^2+\frac{2B^l}{A^2-4B}\left(1- \frac{(1-B^m)(\alpha^{4m}-B^m)}{(\alpha^{2m}-1)^2}\right)$.

(ⅱ) If we take $h = 2l-1$, then it follows from (3.26) that

where

Let $C_{m, l}^{\prime} = u_l u_{l-1}+ \frac{B^{l-1}}{(\alpha-\beta)^2}\left((\alpha+\beta) -\frac{2(\alpha-\alpha\beta^{2m})(\alpha^{4m}-B^m)} {(\alpha^{2m}-1)^2}\right)$. Then

So we have

where $C_{m, l}^{\prime} = u_l u_{l-1}+ \frac{B^{l-1}}{A^2-4B}\left(A -\frac{2(\alpha-\alpha\beta^{2m})(\alpha^{4m}-B^m)} {(\alpha^{2m}-1)^2}\right)$.

□

3.8.   The proof of Theorem 2.8

In this subsection, we will provide a proof of Theorem 2.8.

Proof. By (1.2), we have

where

Then we have

From (3.2a), (3.27), and (3.28), it follows that

Let $n$ be a positive integer. Then it follows from (3.29) that

where

Then we have

By (3.2b), (3.30), and (3.31), we have

Then

So we have

□

4.   Conclusions

Let $\left(u_n\right)_{n\geq0}$ be the special Lucas $u$-sequence defined by $u_{n+2} = Au_{n+1}-Bu_n, \, u_0 = 0, \, u_1 = 1$, where $n\geq0$, $B = \pm1$, and $A$ is an integer such that $A^2-4B > 0$. In this paper, we study the asymptotic behavior of the sequences involving $u_n$. In Section 1, we give the definition of the asymptotic behavior and introduce the asymptotic behavior of some sequences. In Section 2, we give the asymptotic formulas for $\left(\sum\limits_{k = n}^\infty a_k\right)^{-1}$, where

$m, \, l$ are positive integers, $s = 1, 2, 3, 4$, and $C$ is any constant. In Section 3, we give the proof of these results.

Use of AI tools declaration

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

Acknowledgments

Hongjian Li was supported by the Project of Guangdong University of Foreign Studies (Grant No. 2024RC063). Pingzhi Yuan was supported by the National Natural Science Foundation of China (Grant No. 12171163) and the Basic and Applied Basic Research Foundation of Guangdong Province (Grant No. 2024A1515010589).

Conflict of interest

The authors declare there are no conflicts of interest.