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On the Lucas sequence equation \(\frac{1}{U_n}=\sum _{k=1}^{\infty }\frac{U_{k-1}}{x^k}\)

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Abstract

In 1953 Stancliff noted an interesting property of the Fibonacci number \(F_{11}=89.\) One has that

$$\begin{aligned} \frac{1}{89}=\frac{0}{10}+\frac{1}{10^2}+\frac{1}{10^3}+\frac{2}{10^4}+\frac{3}{10^5}+\frac{5}{10^6}+\ldots , \end{aligned}$$

where in the numerators the elements of the Fibonacci sequence appear. We provide methods to determine similar identities in case of Lucas sequences. As an example we prove that

$$\begin{aligned} \frac{1}{U_{10}}=\frac{1}{416020}=\sum _{k=0}^\infty \frac{U_k}{647^{k+1}}, \end{aligned}$$

where \(U_0=0, U_1=1\) and \(U_n=4U_{n-1}+U_{n-2},n\ge 2\).

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Acknowledgments

The research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program”.

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  1. Mathematical Institute, University of Derecen, P.O.Box 12, Debrecen, 4010, Hungary

    Szabolcs Tengely

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  1. Szabolcs Tengely
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Tengely, S. On the Lucas sequence equation \(\frac{1}{U_n}=\sum _{k=1}^{\infty }\frac{U_{k-1}}{x^k}\) . Period Math Hung 71, 236–242 (2015). https://doi.org/10.1007/s10998-015-0101-4

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