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Curious properties of generalized Lucas numbers

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Abstract

Let \(\{V_n(P,Q)\}\) be the Lucas sequence of the second kind at the nonzero relatively prime parameters P and Q. In this paper, we present techniques for studying the solutions (x, n) with \(x \ge 2, n \ge 0\) of any Diophantine equation of the form

$$\begin{aligned} \frac{1}{V_n(P_2,Q_2)}=\sum _{k=1}^{\infty }\frac{V_{k-1}(P_1,Q_1)}{x^k} \end{aligned}$$

in both cases \((P_1,Q_1)=(P_2,Q_2)\) and \((P_1,Q_1)\ne (P_2,Q_2)\), where \(Q_1, Q_2 \in \{-1,1\}\). Furthermore, we represent the procedures of these techniques in case of \( -2 \le P_1, P_2 \le 4\).

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Acknowledgements

The author would like to express his thankfulness to the anonymous referee for the useful comments that improve the quality of the article.

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  1. Faculty of Computer Science and Mathematics, University of Kufa, P.O.Box 21, 54001, Al Najaf, Iraq

    Hayder R. Hashim

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  1. Hayder R. Hashim
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Correspondence to Hayder R. Hashim.

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Hashim, H.R. Curious properties of generalized Lucas numbers. Bol. Soc. Mat. Mex. 27, 76 (2021). https://doi.org/10.1007/s40590-021-00391-7

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