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Electronic Research Archive, volume 33, issue 1, pages 409-432

The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers

Publication type: Journal Article
Publication date: 2025-01-24
scimago Q2
SJR: 0.385
CiteScore: 1.3
Impact factor: 1
ISSN: 26881594
Abstract
<p>Let $ \left(u_n\right)_{n\geq0} $ be the special Lucas $ u $-sequence defined by</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{n+2} = Au_{n+1}-Bu_n,\quad u_0 = 0,\, u_1 = 1, $\end{document} </tex-math></disp-formula></p><p>where $ n\geq0 $, $ B = \pm1 $, and $ A $ is an integer such that $ A^2-4B &gt; 0 $. Let</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ 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}, $\end{document} </tex-math></disp-formula></p><p>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</p><p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \underset{n\to\infty}{\lim}\left(\left(\sum\limits_{k = n}^\infty a_k\right)^{-1}-g_n\right) = 0. $\end{document} </tex-math></disp-formula></p><p>For example, we show that</p><p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \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. $\end{document} </tex-math></disp-formula></p>
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