The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

Hongjian Li; Kaili Yang; Pingzhi Yuan; Hongjian Li; Kaili Yang; Pingzhi Yuan

doi:10.3934/era.2025020

Electronic Research Archive

2025, Volume 33, Issue 1: 409-432. doi: 10.3934/era.2025020

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Research article

The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

1.
School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received: 15 October 2024 Revised: 31 December 2024 Accepted: 15 January 2025 Published: 24 January 2025

Let $ \left(u_n\right)_{n\geq0} $ be the special Lucas $ u $-sequence defined by
$ u_{n+2} = Au_{n+1}-Bu_n,\quad u_0 = 0,\, u_1 = 1, $
where $ n\geq0 $, $ B = \pm1 $, and $ A $ is an integer such that $ A^2-4B > 0 $. Let
$ a_k = \frac{1}{u_{mk}^s},\,\frac{1}{u_{mk}+u_{mk+l}},\,\frac{1}{\sum\nolimits_{i = 0}^l u_{mk+i}},\,\frac{1}{u_{mk}u_{mk+2l}},\,\frac{1}{u_{mk}u_{mk+2l-1}},\,\frac{1}{u_{mk}+C}, $
where $ m, \, 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
$ \underset{n\to\infty}{\lim}\left(\left(\sum\limits_{k = n}^\infty a_k\right)^{-1}-g_n\right) = 0. $
For example, we show that
$ \underset{n\to\infty}{\lim}\left(\left( \sum\limits_{k = n}^\infty \frac{1}{u_{mk}}\right)^{-1}-\left(u_{mn}-u_{m(n-1)}\right)\right) = 0. $
- generalized Fibonacci number,
- reciprocal sum,
- asymptotic formulas
Citation: Hongjian Li, Kaili Yang, Pingzhi Yuan. The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers[J]. Electronic Research Archive, 2025, 33(1): 409-432. doi: 10.3934/era.2025020

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Abstract

Let $ \left(u_n\right)_{n\geq0} $ be the special Lucas $ u $-sequence defined by

$ u_{n+2} = Au_{n+1}-Bu_n,\quad u_0 = 0,\, u_1 = 1, $

where $ n\geq0 $, $ B = \pm1 $, and $ A $ is an integer such that $ A^2-4B > 0 $. Let

$ a_k = \frac{1}{u_{mk}^s},\,\frac{1}{u_{mk}+u_{mk+l}},\,\frac{1}{\sum\nolimits_{i = 0}^l u_{mk+i}},\,\frac{1}{u_{mk}u_{mk+2l}},\,\frac{1}{u_{mk}u_{mk+2l-1}},\,\frac{1}{u_{mk}+C}, $

where $ m, \, 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

$ \underset{n\to\infty}{\lim}\left(\left(\sum\limits_{k = n}^\infty a_k\right)^{-1}-g_n\right) = 0. $

For example, we show that

$ \underset{n\to\infty}{\lim}\left(\left( \sum\limits_{k = n}^\infty \frac{1}{u_{mk}}\right)^{-1}-\left(u_{mn}-u_{m(n-1)}\right)\right) = 0. $

References

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