The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914044522266624 |
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| author | Wan, Yongkang Liang, Zhonghao Liao, Qunying |
| author_facet | Wan, Yongkang Liang, Zhonghao Liao, Qunying |
| contents | In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\leq i\leq t \right)$, Theorem \ref{theorem 3.1} is just Theorems 2.1, 2.5-2.6 in \cite{A22} given by Yuan et al. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences Wan, Yongkang Liang, Zhonghao Liao, Qunying Number Theory In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of $\left(\sum\limits_{k=n}^{\infty}\frac{1}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$ and $\left(\sum\limits_{k=n}^{\infty}\frac{\left(-1\right)^k}{\sum\limits_{i=0}^{t}s_{i}W_{mk+l_i}}\right)^{-1}$, respectively. In particular, for some special $a,b,p,q,m,t,s_i$ and $l_i\left(0\leq i\leq t \right)$, Theorem \ref{theorem 3.1} is just Theorems 2.1, 2.5-2.6 in \cite{A22} given by Yuan et al. |
| title | The inverse of the (alternating) infinite sum of the reciprocal of the weighted sum for generalized Fibonacci sub-sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.14569 |