Hashim, H. R.;Tengely, Sz.

Miskolc Math. Notes19, No. 2, 865-872 (2018).

Summary:F. Stancliff, [“A curious property of \(a_{ii}\)”, Scripta Math. 19, p. 126 (1953)] noted an interesting property of the Fibonacci number \(F_{11}=89.\) One has that \[\frac{1}{89}=\frac{F_0}{10}+\frac{F_1}{10^2}+\frac{F_2}{10^3}+\frac{F_3}{10^4}+\frac{F_4}{10^5}+\frac{F_5}{10^6}+\cdots.\]B. M. M. de Weger [Rocky Mt. J. Math. 25, No. 3, 977–994 (1995;Zbl 0852.11012)] determined a complete list of similar identities in case of the Fibonacci sequence, the solutions are as follows \[\frac{1}{F_1}=\frac{1}{F_2}=\frac{1}{1}=\sum_{k=1}^{\infty}\frac{F_{k-1}}{2^k},\;\;\frac{1}{F_5}=\frac{1}{5}=\sum_{k=1}^{\infty}\frac{F_{k-1}}{3^k},\] \[\frac{1}{F_{10}}=\frac{1}{55}=\sum_{k=1}^{\infty}\frac{F_{k-1}}{8^k},\;\; \frac{1}{F_{11}}=\frac{1}{89}=\sum_{k=1}^{\infty}\frac{F_{k-1}}{10^k}.\] In this article we study similar problems in case of general Lucas sequences \(U_n(P,Q)\). We deal with equations of the form \[\frac{1}{U_n(P_2,Q_2)}=\sum_{k=1}^{\infty}\frac{U_{k-1}(P_1,Q_1)}{x^k},\] for certain pairs \((P_1,Q_1)\neq(P_2,Q_2).\) We also consider equations of the form \[\sum_{k=1}^{\infty}\frac{U_{k-1}(P,Q)}{x^k}=\sum_{k=1}^{\infty}\frac{R_{k-1}}{y^k},\] where \(R_n\) is a ternary linear recurrence sequence. The proofs are based on results related to Thue equations and elliptic curves.

Keywords:

Lucas sequences;Diophantine equations;elliptic curves