Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences

CJM Vol. 2 No. 1 (June 2010), pp. 93 – 100.


Abstract: 

Let $\{U_n\}^∞_{n=0}$ and $\{W_n\}^∞_{n=0}$ be two sequences defined by $U_0 = 0$, $U_1 = 1$, $U_{n+2} = pU_{n+1} + qU_n$ and $W_{n+2} = pW_{n+1} + qW_n$ ($W_0$,$W_1$ arbitrary) with $p, q \in \mathbb{R}$; $p^2 + 4q > 0.$ The aim of this paper is to prove
\[ \sum_{n=1}^N \dfrac{(−q)^{at−1}U_{at^n(t−1)}}{ W_{at^n}W_{at^{n+1}}}
=
\sum_{n=1}^{\frac{at^{N+1}−at}{2}} \dfrac{(−q)^{at−1}p}{W_{at+2(n−1)}W_{at+2n}} \\
= \dfrac{1}{W_0W_2 −W_1^2}\left( \dfrac{W_{at−1}}{W_{at}} − \dfrac{W_{at^{N+1}−1}}{ Wat^{N+1}}\right), \]
where $a, t \in \mathbb{N}$ and $t \ge 2$ . This identity generalizes a number of known identities such as $\sum^∞_{n=0}\dfrac{1}{ F_{2^n}} = \dfrac{7−\sqrt{5}}{2}$ , where $\{F_n\}$ is the Fibonacci sequence.


2000 Mathematics Subject Classification: 

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